I suppose it varies by organization, but I remember the big no-kill shelter here has a formal process for volunteering, with an orientation class, etc. The idea that an outfit like that isn't too interested in having people wander in and pet cats for hours is just a hunch, though, not a confirmed truth.
I don't want to start a debate as to whether or not a highly structured outfit like that is a good idea, just was pointing out that this is something you might run into if you were to check with shelters in lieu of a cat cafe.
I wonder if enough NYers would pay enough to go to a cat cafe to cover the rent? In Japan they are just so nuts about cats that even (what I presume are) extremely high Tokyo rents work somehow. Somehow I'm not sure people in the US are quite as interested, but then, you never know. I do hope someone is trying out this idea.