cherry pie, congrats!!! MOMA would be a fantastic opportunity and after MOMA, there are so many venues in NYC that you might want to stay forever! The connections made at MOMA are probably something you want to think twice about before giving up.
That said, living in NY is hard. Prices are steep and space is cramped. Walls are thin. Buildings are old and dirty. Yes, it's a warning.
My first suggestion is, DON'T DON'T DON'T pay for an apartment listing service. These "businesses" have listings in all the major newspapers with amazing spaces for really low rent. You inquire about an apartment and they tell you they have a list of many more, if you'd just pay for the list. It might run about $200. In return, you get a fake list. BEWARE!
Lots of ppl find success with Craigslist. There are apartments, and roomies, which you may want to think about. If you consider doing the roomie business, make sure your screening process is great or you have a high tolerance for drama.
As for $... It really depends on where you live. A studio might run you $900 - $3,000+. NYC has a great subway system, so you may consider living in the outer boros for cheaper rent/larger space. In that case, I suggest Brooklyn or Astoria, Queens. Long Island City might be a place to consider too. Astoria and LIC will put you 15 minutes away by train from midtown Manhattan. You might find a studio or 1 bedroom in either place or in Brooklyn for about $850, or a roommate situation for around $750.
Ohhh... one more thing: brush up on your cooking skills. There are plenty of places to dine in NYC, but if you're truly on a student's budget, you may want to save the $ and cook yourself, as it can get VERY expensive to dine out.