Well, at least you've got the right attitude.

However, be sure she's onboard with it. Women may wear impractical clothing and shoes to a date, and that could screw the experience for her.

I suggest not packing a lunch, but hitting a deli counter at some "name" deli and splitting a sandwich (which are too big for one person anyway). You'll need to be near a bench or some other place to eat it. Advice from hiking: Bring a lightweight 30-gallon trash bag, which you can spread over a wet bench.

Museums can be cheap, and there are some interesting ones there -- about the only justification I know of to visit NYC.

A show can be good, but TKTS is a gamble and can eat up your day waiting. I prefer not to bother.