The Persian Empire was weakening for generations before Alexander turned up, more and more they were dependent on mercenaries to fight their wars (mainly Greek mercenaries at that), their dynastic politics were absolutely toxic and the satraps were growing increasingly independent-minded, and there'd been a series of major revolts already.
I would bet that the Achaemenids had maybe another generation in them... But that doesn't necessarily mean the end of the Persian Empire. The main thing that ended the Persian Empire in OTL was not Alexander's conquest, it was his generals squabbling over the empire after his death. It's quite possible to imagine an Acheamenid Shah managing to reform the empire, a usurper claiming the Acheamenid name and reforming the empire (it had already happened once before), an external conqueror coming in and taking over the empire but arranging their succession better than Alexander, a Greek (or other) mercenary taking over the empire from within...
The area between the indus river and the sinai peninsula seems to have formed a fairly natural imperial unit, so while Persia might never have too secure a hold on Greece and Egypt, it is fairly plausible to imagine a situation where Persia becomes an eternal empire like China (or rather, an eternal empire on a larger scale than OTL's Persia).
fasquardon