It sounds like it would be a great opportunity for you.
I'm hearing two major concerns right off the bat from your posts.
1. Not knowing many people in NYC and not having anyone there to live with.
- I would say take the risk. You said that you know a couple people so they could help you get on your feet and find your way. When you are working in a university/research setting, it's fairly easy to meet people, at least from my experience. Moving somewhere completely different is very difficult. I went through it for my Masters (moved to the other side of the world not knowing the country or a single person) and am now preparing to move across the country not knowing the city I'm moving to or anyone in it. From my experience doing this for my Master's, I learned a hell of a lot about myself and definitely grew as a person. While it was intimidating and lonely at first, I wouldn't change it for the world. I now know what I'm made of, believe in myself more and know that I can make friends quite easily. Honestly, I would suggest flying the coupe and your comfort zone to 90% of the people I know (the other 10% are totally dependant on one thing or another, for example their parents, so they wouldn't have a chance lol). What helped me through this, and what is keeping me sane right now before my next move, is that you can always go back home. There is nothing written in stone that if you give the job 3 months (so even if you hate it you stay) and then decide it's not for you, you can't pick up and leave.
2. That you really wanted to have some time before jumping into such a big commitment and don't want to rush into something you may not like.
- I agree with the other poster who said that you should see if you could start in a couple months so you can firm up your moving plans first. Then you can take some time and relax a bit and ponder life. A break right after finishing your degree is usually a really healthy thing. As for rushing into something that you may not like, I don't mean this to sound cold, but that's life. I don't remember starting one job without the worry of 'is it going to suck'. You will never truly know unless you try it. You may have the idea that you are going to love it, and get there and find out it's totally not what you expected and the people you work with are horrible. Or.. you may get there with low expectations and feeling pretty negative and then discover that that is truly where you want to be in life and it's the perfect job for you. In the end, regardless of which way it turns out, at least you will have some more experience to guage the direction you want your life to take. Is it still in neuroscience? Is it med school or a Ph.D? Do you like the NY life or would your rather be closer to your home town?
Ok, to sum up this long winded post (ya, I'm bored), I would go for it. It sounds like a gift to me actually. The RA job is in a field you really enjoy (maybe not the exact thing, but in the area), the pay is decent, it's a 2 year contract which will give you time to really bond with your supervisors (think references here) and get a load of experience and you would probably get at least a couple co-authored pub's out of it which is great for grad school app's. You can't ask for a much better opportunity straight out of undergrad. The few con's that there are (based on your posts) can be worked through. You can make friends and learn the city and hopefully push your start date back a bit. Furthermore, it you hate it entirely you can quit and go home. So ya, I'd totally say go for it.
Regardless of what you decide though, congrats on the offer. Let it boost your ego cause I think this is a rare opportunity to be offered.
Anyway, again that's just my own experience but just wanted to toss that out there that it definitely can be intimidating and/or scary but it can also turn out really well when you least expect it 
