Former NYU grad here (Partially funded MA). As everyone said already, NYC is of course expensive, but livable if you can cobble together something that works for you. I worked two part-time jobs during my MA and lived with my partner in a below-market value apartment. As far as apartment-searching: start looking at places now to familiarize yourself with how much rent is in certain neighborhoods. Most of my student friends are beginning to live in Queens over Brooklyn, so check those areas out as well.
Regarding health insurance and healthcare: the NYU Health Center was a relatively decent place for primary care, so I used that for physical health. Mental health is a whole different beast in NYC: most doctors can be extremely selective in what they take as insurance, if anything. If access to mental healthcare is important to you, you’ll find a lot of fantastic doctors based in the NYC area, but their affordability varies. I have run across quite a few doctors who have student rates, however.