There isn't a specific branch of mathematics you need to know in order to learn Category Theory. Its main role is in providing a way of describing other parts of mathematics (similar to the way set theory and certain logics are commonly used to formulate definitions and proofs). In other words, it's a kind of 'foundation,' so there isn't really anything under it. (This is my understanding anyway—please correct me if I'm wrong here.)

That said, it will probably be rather difficult to comprehend if you haven't been exposed to a good amount of other work in pure mathematics first. If you wanted one other more introductory subject, I'd recommend checking out Abstract Algebra (lots of other mathematics are expressed in its terms, also; so independent of Category Theory it would probably be time well spent. https://en.wikipedia.org/wiki/Abstract_algebra).

The purely mathematical subjects that are deemed important for programmers are parts of the so-called Discrete Mathematics. The Wikipedia article on it lists what is included. It is a vast collection of topics, but discerning what would be more important (or interesting) for you should not be hard.

Depends on what you do. I work in robotics, so probability and linear algebra are the two most indispensable subjects I know. Graph theory might be a little more generally applicable.

At the end of the day, you don't need a lot of math to write CRUD apps. The more math you know, though, the more approaches you'll have to formulate and solve problems.