Puking up a hairball occasionally is part of the definition of a cat. It will happen, deal with it.
Anti-Hairball Steps ranked for effectiveness:
- Brushing or combing (LOTS of this)
- Soy Lecithin containing food.
- Other stuff like shaving, anti-hairball goop you feed them. (Vaseline works just as well, is cheaper, and won't stain like the other stuff will. All those goops try to make the hair slide right through.)
The cat probably associates the brushing with pulling and pain. Figure out how to brush without that. Get
THIS kind of brush. Specifically; curved metal tines on flat in one direction and curved in other direction, with a handle that allows pulling along the skin. DO NOT "roll" the brush to pull away from the skin like you would with a human. Pull gently starting at the top of the head. Put a brush nearby any place the cat comes to socialize (bed, couch, computer chair). Keep a regular comb in your pocket and GENTLY use that too. Cats can learn to like being brushed if you make it a gentle caressing type thing over "yanking fur out" thing.
Brush when you want the cat to leave your lap anyway. That way you get a little brushing in and can get up for whatever reason.
Deal with matting as it comes up. How to remove a mat: With scissors that has a point... slide the open scissors through and under the mat. Check for loose skin that might be bound up in it, and if none, cut. Repeat. Eventually you will have the mat out, or it will be a patch of short hair that isn't matted. Do not cut across WITH the skin surface. Think of it like cutting a bandage or cast of an arm, you slide the cutting tool under the bandage perpendicular to the skin and cut.
Deal with the mats before doing much brushing. A cat with fur as long as pictured is bound to have some.
Soy lecithin is a protein that helps prevent the mucus collecting the hairball together in the gut. The result is smaller hair chunks go right through instead of collecting where they cause puking. It will also change the hairball from being a small tight bound up "mouse" type ball to looser and longer that's easier to puke up. Look for "anti-hairball" on the food package, then turn it around and check ingredients. Foods that say "anti-hairball" and are not containing that protein do not work.
Spring and fall is when cats shed the most. Get on top of the brushing early in spring (March) and then when it cools off (September usually but could vary.) If the cat spends a lot of time outside the seasonal changes will happen faster, but also much of the puking will happen outside so isn't a big deal.