The best kind of pet that doesn't cause allergies is one that doesn't shed much. It's the dander people are allergic to, not the fur. Good brushing and vacuuming/sweeping will keep allergy problems down.
Many people think hairless cats will help because well, they don't have hair! But this isn't necessarily true.
My Siamese sheds a lot. Every time he's on my lap he sheds. Our longhair oriental didn't shed much imo. The Maus do and our grey kitties shed a lot.
I've read these aren't too bad for allergies.
Just remember, good brushing and cleaning will keep dander down.
Allergies will usually be pretty quick to flare up. Watery eyes, runny nose, etc.
Going to a shelter will likely cause his allergies to flare up but if he can wear a dust mask while looking at cats, that should help.