Well...
For starters, there is the question of how Cyrus emerges victorious. The easy solution is to have the charge of Cyrus and his elite warriors on the centre of Artaxerxes' army resilt in his brother's death; the Artaxerxian army, demoralised by this blow, disintegrates and flees. Cyrus, at the head of his army, continues his march, enters Susa and is proclaimed Great King.
In this scenario, the war is over by 400-399 B.C. However, things aren't easy for Cyrus, as the successful "coup" of his has destabilised the empire. In Asia Minor, Tissaphernes is most likely still in control of the Lydian satrapy and he knows that Cyrus will come for him soon (IIRC, he was the person who accused Cyrus of conspiring against Artaxerxes shortly after Darius II's death, which in turn resulted in Cyrus' imprisonment and Cyrus' decision to claim the throne for himself); therefore, he would probably try to prevent this from happening. In Cyprus, Evagoras could start thinking that this would be an ideal opportunity to finally unite all the city-kingdoms of the island under his rule. In Egypt, the turmoil and the uncertainty in Susa might cause Akoris' rebellion (or its TTL counterpart) to happen sooner than OTL. Last, but not least, his relations with Sparta would most likely change, as now he isn't the usurper who needs all the support he can secure in order to win the throne, but the ruler of the Achaemenid empire and thus, tasked with protecting its interests, which may be opposite to Sparta's plans.
In the short term however, the bond between Susa and Sparta would remain fairly strong, as Cyrus needs to stabilise his realm first and foremost. Therefore, we might see the Ten Thousands or another body of Greek mercenaries continue serving under Cyrus and help him defeat uprisings and rebellions in the empire, particularly that in Egypt. Another reason is that his enemies inside the empire (Tissaphernes, Akoris, Evagoras) may unite in a coalition like the one Evagoras created IOTL in order to bog down the Persians, so he would need the Spartans in order to keep the first and the third occupied (by supporting the cities of Ionia and sending their navy in Cyprus respectively), while he deals with the second. This in turn could result in the rebel rulers and satraps (assuming that the original three have been joined by other notables opposing Cyrus) adopting Artaxerxes' OTL policy vis-a-vis Greece and start supporting the anti-Spartan groups and cities west of the Aegean in order to force Sparta to pull back and keep it busy. Konon, who would most likely remain in the friendly court of Salamis after the defeat at Aegos Potamoi, would be a key actor in these plans, allowing the rebels to come in contact with the anti-Spartan faction in Athens, which would yearn for an opportunity to rid itself of Spartan overlordship and restore its hegemonic position in the Aegean. Other potential allies could be Elis, which might manage to avoid the wrath of Sparta ITTL, as the latter is already with its hands full in the eastern Mediterranean and in Asia Minor and therefore might not want to risk a conflict in the Peloponnese - at least for the time being, Argos (a historical anti-Spartan force in the Peloponnese), Corinthus, Thebes and Boetia, and many cities in the Aegean who want to regain their autonomy and oust the Spartan garrisons.
Now, Cyrus' victory would most likely lead to a proliferation in the use of Greek mercenaries, since they would most likely be considered as the key factor that helped Cyrus win his throne. The rebels would try to tap the potential for recruits in many anti-Spartan cities in Greece. Although the Spartans would most likely impose bans on the various cities, in order to prevent such a development, it would be fairly difficult to prevent thousands, if not tens of thousands of poor citizens deciding to take the opportunity of profitable employment; in Athens, in particular, the authorities might support this unofficially, in order to reduce the financial burden of the thousands of thytae and repatriated cleruchs on the city's limited resources, but also to assist - indirectly - in the efforts to wear down Sparta. In fact, the mercenaries might just be the most dangerous weapon in the hands of the rebels; assuming that Tissaphernes manages to build up a large mercenary force, he might be able to face the Spartans and their allies in Ionia and Lydia, and, perhaps more importantly, defeat Pharnabazus and take control of his satrapy as well. If this were to happen, Cyrus would be cut off from the Aegean and therefore, his Spartan allies and the mercenary recruitment pool, which could potentially put him in a tough spot. This also spells big troble for the Spartans, as they are now virtually unable to get support from Cyrus - particularly financial assistance. This would most likely force them to start cutting their forces to save money, particularly in the navy, since there is no direct naval threat and potential developments in Greece - perhaps the Spartans decide to restore their infuence in Elis, especiall if it has started to drift to the anti-Spartan bloc or there is trouble in central Greece (between the Phokians and the Boetians or in Thessaly, where the Alleuades could try bring Pharsalus and Pherai under their influence and thus threaten Spartan interests in the area) would force Sparta to raise more armies. This in turn could offer the rebels an opening to swiflty destroy Spartan hegemony in the Aegean, which in turn would isolate the Spartans and their allies in Asia Minor, forcing them to retreat. Konon would definitely press for such a strategy as, again, it would benefit Athens in the medium and long term. Now there is the question whether the rebels could afford both a major naval buildup in order to be able to pursue such a strategy and also the large mercenary armies they would field; I think that, judging from the sums involved in the OTL naval programme before 395 B.C., the three major allies would be able to pay for all this, although it would be a bit more difficult than it was for Artaxerxes.
Assuming that the allies manage to create a navy large enough to be able to face the Spartans and Konon is put in command of it (I think that he would be the admiral ITTL as well, he would most likely be the most capable and dependable naval commander they would have readily available), then things for the Spartans would look bad. However, since Lysander would most likely be still alive and the situation would seem critical enough - especially if Konon has already managed to score a naval victory and cities in the southern Aegean start defecting to him, the Ephoroi may overcome their reluctance and entust him with the command of the fleet. This in turn would pose a threat for Konon, since Lysander was most likely a far more talented naval tactician than Peisandrus - and it would be an interesting showdown, considering that Konon and Lysander had already faced each other in the battle that sealed the fate of Athens in the Peloponnesian War. However, there is the possibility that Lysander, who had grown somewhat old and may be prevented from doing his job by the various controls and checks the Ephoroi and Agis would have put in place in order to prevent him from rebuilding his power base in the cities of the Aegean or exploit his service there to his own ends (although not having the ability to restore the Dekarchies and his local allies would be beneficial, as it had been these regimes that had caused the surge of anti-Spartan feelings in the area in the forst place), as well as lack of support from Sparta - both due to the financial constraints and the fear of the Spartan authorities) and some unpopular measures would create troubles in Spartan fleet (perhaps crews aren't paid enough and they rebel or refuse to serve) and with the locals (which would result in even more cities defecting to the enemy); Konon, for his part, whould probably be determined to defeat Lysander, both to open the Aegean to his fleet, but also to wash the defeat at Aegos Potamoi and he would prepare as best as he could to achieve this. It is possible therefore that, even with Lysander's better leadership, the Spartans would lose the battle for the Aegean. If Tissaphernes was able to better coordinate with Konon, they would most likely manage to expel all the Spartan forces from the Aegean and therefore be more successful than OTL (no Derkilydas in Sestus and Abydus and no Spartan garrisons in Byzantium and Lesbos remain), thus completely cutting of the enemy armies in Asia Minor.
This would be very bad news for Sparta; with Tissaphernes controlling most of Anatolia and now undistracted from the Aegean, he could concentrate his efforts in defeating its forces in Asia Minor; in Greece, the anti-Spartan cities, encouraged by the recent Spartan defeats and the fact that a large part of its army and that of its allies is isolated across the Aegean (especially the Spartans, due to Cyrus' generous financial assistance in the beginning - he would want to cause as much trouble to Tissaphernes as possible - sent a considerably larger force than OTL there) could move swiftly in order to exploit the greater Spartan weakness. The Spartans then might react to this by moving to eliminate any potential threats in the Peloponnese - particularly Elis, but also Argos - before the coming war. This however could backfire spectacularly and find Sparta facing an enemy coalition in its home area for the first time in more than 20 years and with reduced military potential. Such a development would also encourage the Boetians and the Corinthians to sever their ties with Sparta and the former to start attacking Spartan allies in central Greece, in order to destroy Spartan inlfuence in the area. Forced to deal with more pressing matters close at home, the Spartan leadership would most likely be forced to focus entirely on the Peloponnese, allowing the Boetians to eliminate pro-Spartan groups across the Isthmus, bring Phokis in its sphere of influence, alongside much of central Greece and evict the Spartan colonies and garrisons in Malis and Thessaly.
In Athens, the city would jump to join the anti-Spartan camp as soon as possible; they would probably send support to the anti-Spartan cities in the Peloponnese. Konon would most likely sail there, in order to give the Athenians funds to speed up the reconstruction of the Long Walls and the walls of Peiraeus and finance their military buldup. He might be able to convince Tissaphernes to also support the creation of a powerful Athenian navy, in order to hand over the responsibility of containing the Spartan fleet to Athens and concentrate more in his fight against the Spartan forces in Asia Minor and later on, Cyrus. therefore, the Athenians could start laying the foundations for the reestablishment of their influence in the Aegean; also, unlike OTL, the fact that the Athenians woudn't face siginificant threats from the land at least for some time, as central Greece would be fully under the control of the anti-Spartan coalition and Sparta would be badly pressed in the Peloponnese, would mean that they could dedicate even more resources in naval rearmament and also, not tax their economy way too much, which in turn would allow for the private economy and the city's finances to recover more quickly than OTL and consequently, allow the Athenians to continue the war effort more easily than OTL and also fully turn their energies in securing their areas of interest in the Aegean and the Hellespont.
Once the new Athenian fleet had been created, they would most likely start raiding the coast of Laconia and Sparta's allies in Argolis; but the main focus would probably be Aegina, as it was a Spartan naval base very close to Athens. Due to the aforementioned factors and Sparta' greater problems, the Athenians could succeed in conquering the island and neutralising the threat it posed. After that, there would be expeditions to restore the cleruchies in Skyros, Imbros and Lemnos (perhaps they would try to establish another one in Aegina, although the problems this would create with Megara and Corinth might dissuade Athens' leaders from making that decision) and efforts to recreate a web of alliances that would allow the Athenians to reestablish some sort of influence and control over the cities of the Aegean, the Hellespont and Bosporus (perhaps some sort of an earlier and looser second Athenian alliance).
Sidenote: If these events were to happen in 398-397 B.C., then these development could also affect the West: with the Spartans more badly pressed than OTL, they may be unable to send reinforcements to Dionysius of Syracusae as they did IOTL with Pharax. This would mean that his fleet would be somewhat weaker than OTL, which in turn might allow Himilcar to intercept the mercenaries he would have recruited in the Peloponnese (assuming that he sent Polyxenus there for that reason ITTL as well). This would make the siege of Syracusae last longer than OTL; but more importantly, if the Carthagenian army still succumbed to illness (especially if they had camped near the swamps), Pharax's absence and perhaps Corinthian support during the siege could make the current of anti-Dionysian feelings more powerful and allow his oppoents to organise better, thus allowing them to overthrow him.
In Asia Minor, if Cyrus is bogged down in Phoenicia (which might have also revolted against him and Egypt (especially if Akoris or his TTL counterpart have managed to recruit large numbers of mercenaries and have secured the services of a capable general (Iphicrates would be a good option, as he could be available), Tissaphernes would be free to deal with the Spartan army as he saw fit. It wouldn't be easy but in the end, short of reinforcements and deep into enemy territory, Agesilaus (or whoever the commander of the force was) would be forced to retreat. In this case, the Spartans and their allies would probably sustain larger losses, as they would have to fight their way to the sea, and then they would have to face the Athenians blocking the way to Europe in the Hellespont and Byzantium. Furthermore, even if they managed to cross they would have to face Thracian raids, the Athenian fleet in northern Aegean, perhaps attacks of the Chalcidean koinon, Meidius and much of the Thessalian koinon and the Boetians and their allies as they travelled back home by land. Therefore, the Spartans could ultimately receive a weak and decimated army, which wouldn't be enough to reverse the situation to their advantage. However, it is also possible that the army be destroyed in Asia Minor.
In both cases, the Spartans would most likely seek for peace after some time. Although at first glance this would seem very difficult, considering the allies would be in a very advantageous position, the divisions in the ranks of the latter might offer an opening: after having reclaimed their hegemonic role in the Aegean and become once again a naval power, Athens may consider dropping from the war, especially if the Boetians start to look menacing (and there could be causes of friction between the two, such as Oropus and Plataies and the Spartans are ready to accept the new status quo at sea; the Boetians could accept an end of hostilities if their sphere of influence in central Greece was recognised; the Elians could consider peace if the Spartans allowed them to remain outside their alliance and keep Trifylia; and the Corinthians would want to end the war, as it would most likely prove to be a rather expensive affair (especially when one considers the fact that the city was financially ruined by the Peloponnesian War) and Argean ambitions could make the more moderate elements in the city anxious. Therefore, if enough cracks began to appear, the Spartans managed to score a victory decisive enough to make the allies doubt whether they could bring down Sparta entirely and funding from Tissaphernes and others dried up, the two sides might end up at the negotiations table. In that, the Spartans would have to make serious concessions, such as allowing Elis and Corinth to leave the alliance, recognising the Athenian sphere of influence in the Aegean and the Boetian gains in central Greece.
Although this would seem like a major victory for the anti - Spartan forces, things wouldn't so clear-cut: Corinth would probably approach Sparta after the war due to the fear of Argos; Elis too could have a rapprochement with Sparta, especially if the aristocratic faction managed to come to power; Megara would certainly have troubles with the Athenians, which could offer Sparta an opening there. Outside the Peloponnese, the Thessalians, particularly the Alleuades, might start growing anxious over the power and the Boetians (especially if they were to start supporting democratic groups in the area) and seek Spartan support; the same could happen with Macedonia, this time in order to deal with the Chalcidean koinon; and in the West, Spartan diplomacy might manage to court the support of Syracusae or Dionysius - if he has managed to remain in power. It wouldn't be easy, but if the Spartans managed to carry out enough reforms to regain most of their previous power or at least the perception of it - and with some lack, they could pull this off. Furthermore, if Athens or the Boetians grew too strong, Sparta might manage to be the decisive factor in the balance of power beyond the Isthmus, which would be a rather good deal for them, considering their setbacks.