My standard procedure in cat cafes is to hang out until the cats decide to bother me. It helps if you bring a book or laptop, just don't plan on getting to use them.

Tinker's Cat Cafe in Salt Lake City

But even just a phone works. One time at the San Diego Cat Cafe, a cat decided I was spending too much time looking at my phone and climbed into my lap to get to it, while brushing my face with its tail like a swiffer brush.