Renting apartment in NYC?

No you do not need a greencard to rent but you will not be able to rent directly from a landlord for such a short time. You can airbnb and then find a sublet on Craigslist. They have some scams which is why you cannot do that remotely but they are a good way to find a short let. For example you can often find a room where one of the tenants had to move out early.

If you are going there for an internship, they might help you find something.

no, but most rentals are for longer than that like a year, so you might have trouble finding one. A residential hotel might be your best option., like http://www.residenceinnnewyork.com/manhattan-suites/

Most landlords will not do that short of a rental. You need to find a vacation rental or extended stay hotel.

you are not likely to find an apartment in NYC to rent for two months.. try a hotel