As a melee player who has played ultimate I can say you're doing something wrong. Either you're holding the controller too tight, or something else. You want do to a few things.
1. You want all your hand motions with the controller to be as optimized as possible, no extra strain on the tendons, no needless exertion of force.
2. You want to do hand exercises, bend your hands back and forward, stretch your palm out etc.
3. As soon as you feel any hand pain at all you should stop and if possible put on a brace (it's like a glove with a metal rod in it that restricts hand/wrist movement). If you feel pain on a day where you don't play at all, you're not playing that day, you're wearing that brace and resting your hand and wrist.
If you want to keep playing for the rest of your life (like most melee players do, atleast) you gotta be careful and catch this when it starts. Is it overly cautious? Yeah, but we're trying to be preventative. Even at Smashfests you gotta remember people rotate out, people drive to restaurants to get some snacks, people have beers and laugh with eachother. 12 real active consecutive hours is a rarity.
