Hello! Good on you for getting your bunny neutered, he might be fussy about it now, but he’ll appreciate you later.
Cut off as much jumping source as you can for at least a week, but it seems you’re already considering that! Lots of people build their bunnies small recovery cages that work pretty well while they rest. You can give him access to an area to play in as long as its mostly flat, but I’m not too sure about the cat thing, myself.
I know you’re worried about upsetting his tummy, but rabbits will usually take medication when it’s disguise. Your bunny will be very uncomfortable after surgery, so for his best recovery, it’s advised to definitely give him pain meds. If you’re not fond of crushing up a banana, both my bunnies get antibiotics a day, and I feed them a raspberry before meal time each. Both don’t seem to realise too quickly that the hole in the raspberry contains their medicine. This is pretty much the only time they get raspberries, but it is a bunny safe berry, and one won’t hurt. 