Congrats on the volunteering - good for you.
NYC is full of free entertainment. Take advantage of that. Some great stuff happening.
Health care: New York has a free health care plan for the poor, but if you're living in NJ, I think it's only open to kids these days and isn't nearly as good.
Barter works best with alternative providers - help out your local acupuncturist or chiropracter, get free health care. I don't think it would work as well with regular MD's. Chinese medicine has been around a lot longer than US healthcare, try to find someone in that line - and haggle for the best price.
Also, see if you can stock up on medicine you might need while you still have a job and prescription plan - antibiotics, painkiller, allergy, whatever you can think of.
And don't do stupid stuff like jump off buildings for a dare, or try and beat a bus across 42nd street.
Dental I haven't any ideas, except to brush and floss regularly, and get any necessary or preventitive work done while you still have a dental plan.
Exercise: this might be a great time to become an expert on stair-climbing. NYC has lots of stairs. There are groups that climb the stairs in tall buildings. Also, there are all sorts of things like races that you can sign up for and have some fun - and of course, collect t-shirts. You can often haggle the entry fee - and lots of group things have snacks etc. Walking across the Brooklyn Bridge, Geo. Washington Bridge, etc is one of the great experiences in this world. And so on.
One way to get a good workout is find a gym and ask if you can teach a class, or do some one-on-one coaching, for free, and in return get use of some of the facilities.
Food: Get to know the food pantries and the soup kitchens. I used to volunteer AND eat back when I was in college. Some amazing people there.
Breakfast is the cheapest meal to eat out - look around for specials. Also, look out for coupons, specials, whatever for free food. Let your friends know that you'd appreciate leftovers or food they have around the house that theyre going to throw out when cleaning, or stuff they get for presents that they won't eat. And, find out who's allergic to stuff and take things off their hands.
The chinese restaurant across the street has a takeout bowl of wonton soup for under a dollar, and they throw in those crunchy noodles, cookies and teabag - and if I ask nicely, they'll give me extra noodles, cookies & teabags. Just a hint, find out what the cheapest thing on the menu is at your local spots. And find out what places throw stuff out at the end of the day/night and don't be shy about begging. My local bagel bakery gives away free bagels right before closing, for example. I'm not in NYC, but you can find this stuff anywhere.
And, don't be shy about asking all of us ATOT'ers to invite you to dinner.
The hard part about cooking cheap meals is that you often overdo it on the cheap starch and skimp on the veggies - and that will affect your health. Veggies can get expensive, but you need them. 5 decent servings a day of veg/fruit, that adds up on a weekly basis, but don't let it slide. Also, stock up on vitamins while you still have a job.
And, the traditional way to eat cheap in artsy NYC is to get on lists for openings at art galleries, book parties, movie openings etc. Wine-and-cheese is standard. Sometimes there's a really great spread. Don't be shy about asking people for invites to these things.
I've heard of people doing things like going to funerals for the food - people always get together to eat afterwards - but I've never done it. I suppose if you have a nice suit and can master the "I'm so sorry" expression, you could do well.
Incidentals: some of the greatest marketers in the world are in NYC, so work on your resistance skills. If you truly have zilch in the money line, just get it into your head that you can't buy ANYTHING. You will be tempted!
Check the papers or the online shopping boards, like here, for specials at the drugstores. I have tons of toiletries etc that were free-after-coupon. Some of which I actually use.
It's tough to balance shopping around for the lowest price, and wasting too much time.
Oh, and don't hang out with people who spend a lot of money. There's no way to make that work, unfortunately. Be very clear upfront when people invite you out for dinner or drinks that you have NO money. You'll find out very soon who you can have a good (cheap) time with, and who you can't.
Most of the cheap/free NYC places I knew back when I was a poor musician are gone, so I won't bother with that, but there will always be people enjoying the pleasures of frugality, so you'll be able to find stuff if you try.
Best of luck!