It's only worth renting a car in New York City 24 hours a day if you're going to drive around the neighboring states. Otherwise, you'll spend a lot of money, as parking lots in the city are very expensive, starting at $20 USD per hour.
To rent a car without a deposit in New York, you'll need a driver's license (and possibly an International Driver's Permit, ID), insurance, and a credit or debit card.
Important! If stopped by a police officer, stay in your car - you can only get out after the officer gives you permission.