New York City is exactly what you make it. A lot of people joke that it costs $20 just to walk out the door (you're always buying a coffee, stopping by at the bodega for a water, grabbing a meal, etc) but if you really work at it you can live near the city on a surprisingly low budget. Here's what I'd say generally:
General Rules of Thumb For Living
If you want to live in a nice(ish) place in Brooklyn or Manhattan and have less than a half hour commute to your job you're going to pay $1,000 - $1,500 and you'll have roommates.
Nice one bedroom apartments will run you anywhere from $3K - $5K per month. If you want to DM me I can show you my apartment and tell you what I pay for it, as well as answer any other relevant questions.
If you're willing to commute far and/or live in a less nice place you can get away with renting a room for around $500.
Food and entertainment wise, the city is endless. You could spend $100+ on dinner and drinks every night if you want and then spend another $20 - $300 or so going to Broadway shows, Operas, the Ballet, concerts, sports games, etc. Or, you can grocery shop, eat $1 pizza slices and skip entertainment and eat on less than $15 per day.
Long story short, if you want to live comfortably you should plan on spending some money here. If you're willing to live a harder life and not socialize as much (or just crash Happy Hours with free buffets) you can live on less. I know I'm tempted to spend at least triple what I do in other countries when I'm here.