My only idea is that if you can find a place to volunteer they may allow you to stay on the premesis. A homeless shelter may let you stay on in a room if you work for certain periods of time during the course of the day. I'm sure you'd have some down time to explore the city if they were to accept.
If you were interested in exploring NY state, I do know of one place where you may be able to volunteer working with rescued, retired and sheltered animals and they may put you up in exchange for some work..possibly provide food and even a small stipend maybe.
I suggest you come up with a coherent, well thought out proposal and contact http://www.petsalive.com . It's an animal shelter owned by a friend and they always need help. They're not close to the city, but you might be able to earn a enough to cover a bus ticket to and from the city from there for a day visit.
Another option is working with Clean Vibes (http://cleanvibes.com/) Another organization run by another friend. What they do is look for people to help clean up the grounds at various outdoor music festivals through the warmer months.
I don't know if they'll be working any New York Festivals this year, but email them and ask or check out what they have coming up on their website. They'll get back to you. They work festivals in various parts of the US including New York (they're based on the East coast) and it's a great way to hear some fantastic non-mainstream music and do your bit for the environment at the same time.
They provide meals, housing and even stuff to wear!