Alright, here's what you need to do for now. Buy Prime if you haven't already, it's not very expensive and removes chlorine from the water, along with ammonia, nitrites and hard metals. Change the water completely every day, or at least every other day. If you still have the container you got him in, scoop some water from the tank and stick him in there while you take his tank and rinse it out, adding new water and a drop of prime. (If you don't have the container, a small glass will work, just make sure it was never exposed to soap) You could add a few pieces of Aquarium Freshwater Salt to aid in healing with Fin Rot, if you can you should also get Stress Coat+, as it has Aloe Vera in it to help heal the fins faster. I would also really recommend getting a heater, as that is a necessity.
Once you have enough money, you really should upgrade him to AT LEAST a 2 gallon tank. Or if you wanna save some work, get a 5 gallon tank with a filter and hood. (Petsmart has a package for around $35) and a small bottle of Tetra Safestart (SafeStart has the bacteria needed to have a cycled tank which removes ammonia and nitrites) along with gravel, a few SILK plants, and a little cave to hide in. Set it all up, pour water in, add prime, then pour the entire small bottle of Safestart in there. You can now add your betta in the new 5g tank, and only have to do 25-50% water changes once a week.