2 gallons is really minimum to keep a betta in.. 2.5-5+ would be a lot better and a lot easier to keep him healthy and he will most likely live longer. For now, this is how you can take the best care of him..
You don't want to use a filter in something that small. You can't cycle something that small and the flow will only stress your fish out. The filters included in those particular minibows are known to sometimes suck fish into the filter as well.. The filter is probably stressing him out and may be part of why he won't eat.
Also if you have not added the conditioner that is also stressing him out. You need to use conditioner because your tap water is toxic to bettas. There are a lot that will work but I like Prime because it will render some of the ammonia that will build up in your small bowl less toxic. I suggest you use that. You use two drops per gallon of the conditioner - the top should have a dropper or some don't and you can use an eye dropper. If you chose to use the conditioner that came with the kit then follow the instructions on the bottle..
You want to do two times weekly water changes at minimum (one 50% and one 100%), but a lot of people will do two 50%s followed by 100%. To do a water change, use a little cup like a plastic solo cup - this cup must be only for him and hhave never been used with soap or other chems. Scoop him up in this cup (keep him in the cup about 1/4 full of water - it doesn't need to be much because he won't be in it for long) and leave him in the cup while you change his water. To do the 50% use a turkey baster - dedicated only to him that has never seen soap or chems - and drag it through the gravel and try to suck as much of the poop out as possible, in addition to 50% of the water. Use a thermometer under the running tap to get it to be the same temp as the water that is normally in his tank (in your case you say 80 degrees). When the thermometer says the flowing tap is the right temp, fill back up his tank. At this point, add the conditioner (dose for how much water you change - if you change half the water you add half gallon worth of conditioner, If you do a 100% water change dose for the full gallon change). Float his plastic cup with him in it in the new water. Slowly add a couple tablespoons of the new water into his cup every 10minutes for at least an hour. Finally, dump him in gently but try to get as little of the old cup water back into the tank as possible. When you do the weekly 100% you will do mostly the same thing except empty his tank fully and rinse everything in it very well under warm water but never use soaps or chemicals. Once it's fully cleaned/rinsed you can refill it and repeat the cup/acclimate phase.
Are you using a heater and an intank glass thermometer like this to monitor heat? http://www.amazon.com/Marina-Floatin...na+thermometer
You need a thermometer to monitor temperature because heaters tend to not be very accurate. Also the ones that work in a 1 gallon tend to overheat the bowls so you much watch temps carefully. 80 is okay, if that's what it actually is. Your bowl temp will be about 4 degrees cooler than your room temp unless you use a heater, so heaters are needed in houses that aren't kept 80F+. Never assume your tank is a certain temp without a thermometer.
You can feed 3-4 pellets split up into two small feedings (so like 2 in the am and 1 or two in the pm) and one day a week he should be fasted and not eat anything. Watch him eat and if any should fall to the bottom of the tank suck them out with the turkey baster. Don't leave them. They will foul up the water quickly. The Aqueon pellets that come with the kits aren't the best for him - they're not the worst either though. But if he won't eat you can try some others. My favorite are New Life Spectrum Betta. They can be found at petco. They are smaller so you feed more. My boys get 5-8 split up into 2 or 3 smaller feedings a day (so like 3 3 2 or 3 2 2, or 3 2) with one fast day a week.