Give up now!

Seriously - in NYC, ATM access shouldn't be a problem and most places will take credit cards. If your question is how much to budget, that is also very dependent on what you plan to do. Walking down Fifth Ave window shopping is free. Shopping on Fifth Ave can get pricy. Museums and attractions vary, but you can set a rough budget by deciding what you want to do each day.

The bigger issue you face is a teenager and spending money - souveniers, or just in general. Just like any trip to your local mall, most teens will want their own $. At the same time, you probably don't want him/her walking around with a week's worth of cash. Depending on the age & maturity/responsibility level, you might use this opportunity to give your kid a cc (probably a card on your account). I would only hand over a cc AFTER setting a per-day spending limit! And plan to total receipts each evening to be sure the limit is adhered to. Or get a prepaid Visa gift-card or similar debit card. It lets him/her decide what they want without you handing over $$$ every time s/he wants something. (If you go the cc route, let your teen pick-up the tab for dinner one evening. Kids love telling the waiter they are buying.)