If the limping is a sudden thing that just started after getting hit with a door then it could be something like bruising, pulled muscle, or a damaged tendon.

Arthritis generally is a gradual thing and is usually most noticed after the animal has been laying down for a while, jumping up, and climbing stairs.

Either way a visit to your vet is in order because it sounds like your cat is experiencing pain and they will have the right meds to help him feel better. Don't try to give asprin/tylanol/motrin to him. Most of those are toxic to cats.

BTW until you can get into your vet's office, I'd keep your cat's activity to a minimum in case it is some kind of damaged tendon, more activity can cause more damage with those.

Oh and at 12 yrs old it is possible for arthritis to start. Ask your vet about getting some glucosamine for cats (Cosaquin, Dasaquin etc).

Good luck.