I'm upset at my coworker about his cat story
Did you not like his cat story?
No, he shouldn't have told me about it.
It's okay to be upset at your coworker. You should discuss boundaries with him around work conversations. That way you won't have any conversations that make you uncomfortable.
Maybe I will tell him my boundaries but for now I'm going to stop his cat stories anyway.
I am glad that you are doing something about the thing that bothers you. But you should understand that getting frustrated with your coworkers is normal and you have to learn to deal with it.