New York is one of my favorite cities all over the world and believe me, I have been to many places. NowI understand your concerns about the budget simply because I myself was a student some years ago and to tell you the truth New York is far away from the common understanding of a cheap city. This is why it is better to make a research on internet and see which places you should avoid and which you should definitely pay attention to. After all it would be a real pity if you go to New York and you spend your time there only in your apartment without doing some sightseeing. Actually, controversial to the common understanding that NY is a really expensive city, there are places which you should go and you will be able to save some money as well.
Lets start with the accommodation. The good news here is that as you know New York is a city of wide variety of choices and you will have the opportunity to choose among many places and neighborhoods. And believe it or not but finding an Apartment Las Vegas is both easier and cheaper than finding a Los Angeles apartment and Apartment Toronto. So be careful while choosing your place for stay simply because there are many offers. Search and see how the prices are going on the real estate market and therefore choose the best place that fits both your budget and needs. By the way if you want to find a decent hostel you can use this site http://www.hostelhandbook.com/ A friend of mine gave it to me and told me that he used it a couple of months ago when he had to spend 1 week in NY because of some work. I took a look at the offers and some of them were pretty acceptable. Rooms started from 40$ per night.
Next thing on the list is sightseeing. As you know the infrastructure in this region is really highly developed and this is you if you want you can pretty easily visit Philadelphia, Boston and Washington DC by bus. I cant tell you the exact price of the tickets but you can use the official website of Greyhound. Anywayif you want to do some local sightseeing I would recommend you some of the museums. You can start with Ruben Museum of Himalayan art, Neue Gallery, The Whitney, The museum of the city of New York, The Brooklyn Museum which by the way should be combined with a visit to the Brooklyn Botanic Garden.
Now about the foodwell as you know this is strictly individual issue. I mean I dont know what type of food you prefer but if you want to save some money you can eat in the fast foods and they are thousands of them in New York so basically there is no way to stay hungry. On the other hand you may find some decent restaurants on cheap prices ” it is up to you!