Actually in NYC they should have architectural plans on hand. My guess too is probably drywall over plasterboard. Especially if it was built in the 60’s. Can’t really be certain, but if you’re using a standard drill bit you are not getting into masonry.

If your pot rack is holding three thin pots or pans you’d be fine with a toggle wing anchor. If you’re hanging your le Creuset collection you are going to have a bad time without lag screws.

In general, if this pot holder has a lot of screws well spaced out? Toggle wings. If they are hanging from a single or close point? You must use lags or something with real structural power.

Are you willing to post more pictures? That’s the only directional advice I can give without seeing the fixture and conditions.

It costs you $0 to ask, and you could potentially learn something very helpful. Definitely give it a go

I see, sry I read apartment and thought landlord :P Other than the stud finder, i really don't know about the wall, good luck!