How was Alexanders army able to deal with the Persian archers?

[Flavius Merobaudes]

I suppose thats true, although i wonder if rome actually had the manpower to outspam persia (not that they need to, having higher quality infantry)

Based on the sheer numbers and vastness of the Persian Empire, one would have concluded that the Romans could impossibly have outspammed the Persians. But I hope you won't mind me to disagree with you there.
From all sources we have we can be sure that the Persian army of Xerxes was vast. Numbers and estimations differ in quite a wide range, but after all they were not innumerable. With a large proportion of the population mobilized and ready for combat, the Romans might possibly have outnumbered the Persian. Take into the equation the attrition and generally the difficulties an invader would have to face in enemy territory, and you might come up with some clear advantage for the defenders - espacially in a protracted war.
The Persian Empire was a its high point under Xerxes, but it could never afford to direct its force against one single foe over a longer period of time. After all, keeping the empire together as it was, consumed the major part of its resources. Thus, a single and all but large invasion was possible, but nothing in the sense of a total war - or else the empire would have gotten into difficulties. One example for that might be the return of the Shahanshah leaving his general Mardonios behind to manage the protracted war on a smaller scale. The Roman concept of war on the other hand comes much closer to what we might call a total war.

I apologize for speculating too much here, but I take your question as speculative in itself. Most probably there is someone with a deeper insight into the topic who can tell us more.

Edit: It is quite interesting to see, how a conversation that started with "Alexander's army vs. Persian archers" can drift off to an imaginary "what-if?: Romans against Persians almost 2 centuries earlier"...