Regarding the Volga, Russia could project a lot of power compared to Astrakhan, but not compared to the Ottomans. Remember that in this period the Crimean Tatars were raiding the outskirts of Moscow - if the Ottomans had made a strong effort toward Kazan, there's not much Russia could have done about it - not in the 16th c.
But the point is, they tried for Astrakhan - and they failed at least once in the 16th century. The Crimeans could raid but couldn't occupy, and they sure could be beaten by Russian armies, as per OTL. As for the Ottomans and their level of interest in the region - they couldn't even hold on to Azov without random Cossakry seizing it from them in the 17th c.
How would you hold it? Overland via the Caucasus is crazy difficult if not impossible. By sea to Crimea and overland to Volga is almost as difficult. Azov and Taganrog were probably depots for just such an expansion, but they changed hands far too much for comfort IOTL. Further, the Russians can easily make occupation of Astrakhan untenable in the long run by raiding along the rivers, which they were really rather good at. By boat in summer, by horse in winter, if they need to.
The Ottomans would have to rely on the Krym Tatars to do the counter-raiding, which was becoming less and less effective throughout the 16th and 17th c. In fact the Tatars had a tough time facing Russian armies without Ottoman troops backing them.
There are a few openings:
1. Livonian War proper - can't see what there is to gain, really, but maybe you could hurt Moscow's environs some more? And to be honest, they already drew all the strength from the allied Khans that they could for that one. So maybe inconsequential, as an opening.
2. Time of Troubles! - Russia is weak on the Volga, a decent chance to make up for the setbacks in the last century. Already a few settlements along the Volga and Dniepr that could actually be siezed. Not a bad attempt.
2b. Razin's revolt. Alexei is in no position to fight a war, he'll have to acqiesca to reasonable demands. But Astrakhan may already be unreasonable at that point.
3. Prut Campaign - while it'd be interesting to see what would happen to Russia without Peter, it's doubtful that the long-term pattern of Ottomans repeatedly losing to the Russians in Kuban or being unable to project much further than the coast when they
do win would change much. Still, not a bad opening. I could see the southern border return to status quo, maybe greater Cossack-Ottoman cooperation, and some compensation for Sweden. Russia shrinks back to what it was under Alexei Mihailovich and takes another fifty years to regroup, maybe never quite does if there's no new leadership arising.
4. Have them do better in the war of 1735 against Russia and Austria. The Austrians were not performing well at all, and the Russians performed well in the field but overstretched their supply, as was usual for their southern campaigns.
That could see them sieze more of Ukraine, I imagine. I don't know whether Astrakhan is even an option at this point, though, even if Crimea does well in the Caucasus in the interim. This is the latest opening.
After that, Russians will consistently beat the Ottomans as they move their centres of operations south ever more solidly.
It's true that the Empire was a lot stronger, but there's good reasons why they couldn't and didn't do much better than they did in the Steppes. The trouble with investing a lot into invading Russia for Astrakhan is that:
1. Difficult to defend - besides the Russians, there's Cossacks, Nogais, Bashkirs, Kalmycks, Chechens, Georgians, Circassians. None of them will magically cooperate with any central authority. It will be constant warfare far away from centers of Ottoman strength.
2. Astrakhan, Kuban's coast and the Crimea are the only things of worth, really, at the time. Everything else is largely empty and ungovernable.
3. There's hardly any population to tax on the steppes, and I doubt that the Ottomans were any better at settling the Steppes than the Russians at this point in time. Do they have excess population to export?
4. Concentrating on Russia (and that would take considerable resources, more than is compensated by taking in a part of the Caspian Trade) would weaken them against the rest of Europe - which, to a point, happened IOTL.