My younger sibling is allergic to nearly everything (milk, grass, sunlight...), including cats, so I never had one growing up. Got a cat when I got married. Turns out I'm allergic to cats. I still have them, and have gotten a couple more over the years.
1) HEPA filters will help a lot. Also HEPA-level filter for your vacuum so you're not throwing pet dander around while trying to clean up.
2) Cat doesn't sleep in my room. Sorry skitty. Having a "safe room", especially where I spend 7-9 hours a day, keeps the allergies way down (and reduces the amount of floor space I'm responsible for cleaning)
3) Regular vacuuming and lint rollering anything cat can get on. It's a pain, and it's the only way. I also wear a mask while I do this, which is probably overkill, but I do it anyway.
4) Daily claritin. benedryl for particularly bad days. Some people swear by daily Zyrtec instead, but I found claritin had fewer side effects.
5) Always wash hands after petting cat. Cats are covered in catlick, as is anything you touch after touching them.
Exposure seems to have helped, honestly. The longer I have them, the less compulsive I am about cleaning and my flare-ups have reduced dramatically.
Talk to your doc about a full allergy panel though. A few years ago, I thought I was having worse symptoms related to the cat, but found out I have a mold/dust allergy. Went looking for mold and found some behind a dresser and wardrobe! So don't be so certain it's (just) the cat you're allergic to. Good luck!