If I were you, I would just go out and get a 5 gal. tank for $6, an azoo palm filter for $8, and a heater for ~$10. It's all pretty cheap, and you will only have to do water changes once per week. Also, with the temp. set at 78 your fish will be much more healthy and active. Drastic changes in temp. (when you put the bowl in warm water) will stress your fish out and make him prone to disease.
As for getting him to eat, sometimes the stress of being in a new home will make them loose their appetite. But, if it has been a week you are right to be getting worried. He may be lethargic because he is so hungry. I would try freeze-dried bloodworms (they float so if he doesn't eat them you can pick them out) or brine shrimp.
Have you checked ammonia and nitrite levels? If you have only been doing a once per week water change, and there is uneaten food in the tank, they could be quite high, poisoning your fish.