Hi Welcome to Fish Lore
I'm sorry your betta is worrying you. What's his name?
When something isn't right, one of the first things to check is the readings for
ammonia,
nitrite and
nitrate.
Do you have a
test kit?
How old is he?
He could be feeling the effects of living in such small quarters with cool water.
Here's a good link to read:
Welcome and Betta Care Guide - Please Read First!!!
I would suggest getting a 5g. tank with a filter and heater.
Stress Coat, Vita-chem and Garlic Guard will help build his immune system.
Good luck, I hope he feels better soon.