Fizz sounds like he's got a caring owner. I was in a similar situation. I bought a betta who was pale white and suffering fin rot, ick, parasites... anyway, He's doing well now.
Your current one-gallon tank sounds like a good effort and is a great start for him since he's been in a tiny cup, as Bombalurina said. I'm not going to harass you about an upgrade as you already plan on doing that eventually and having a 1gal spare tank for later use is good. While he is sick and in the 1gal though you might think about making the plunge to a 5-gallon and cycling it. That will mean fewer water changed later. Taking care of a fish and having to worry about staying at a friends house or leaving for a weekend and getting a fish sitter you trust to not only feed your fish but change the water as well can get iffy if you're really attached to Fizz- though it is possible if you know someone who has fish of their own.
Basically my point is just to think about what you might want in the future, and buy that so you don't make my mistake. I had a 1gal and realized I was changing the water daily and still worrying about it (I tested test the water daily and even with a filter designed for a 10gal it still got icky fast), went to 2.5 but realized I couldn't do much with it, and bought a 5 to start cycling and planting- and after buying 6 plants I was told the lighting in a 5gal was not great and the plants would not do well in them- so now I have no money and I envy the 10gal tank I keep seeing at the pet store. LOL. I bet if I bought it I'd start eying 20gallon tanks though, so I need to just get low-light plants and be happy if my fish is happy.
Change water daily with your 1gal- even if it seems clean. You'll want to set 45 minutes to an hour aside every evening to do this because cleaning gravel and ornaments can take awhile and because your fish needs to adapt gradually to water changes.
With the water, try some AP salt I think 1/2tsp per gallon, but it might be 1tsp... It's been awhile since I used the salt but it did help my fish. Try buying either a gallon bottle or a 2-gallon water jug so you can treat the tap water and allow the salts to dissolve for a few minutes before cleaning the tank.
When you do get finances you might think about a heater, or just a thermometer- if the temp stays in the high 70s then you're fine with the heat lamp until winter.
Good job on that heat lamp, by the way!
Omega Betta Pellets are great betta food if you can get it since I haven't heard great things about freezedried worms and frozen ones are more for treats than for meals.
Let me think... You have heat, you're going to change the water every day, you know about the food- FILTERS! If he's sick you might try turning the filter off at night and for a few hours during the day so he doesn't have to fight it. They also have fragile fins so it could be too much in that small of a tank. For now, I'd take the filter out. Sorry to say but after buying three different filters I've come to learn that the ones for the smaller tanks are mainly just to get people to buy the tanks. They are working a bit, but think about volume.
I turned my filters output to flow into my one-gallon tank and left it running in my 2.5gal which I had put 4 drops of ammonia in (fish waste, ya see). I did this while my fish was in the 5-gallon tank because he doesn't care for science
I checked on it 6 hours later the water in the 1gal bowl still had about 1/4 as much ammonia as the 2.5gal! So it would take 24 hours for my filter to clean the water if I were to pour the water back into the 2.5 and filter it again. However, with a fish tank you don't simply have your fish putting out ammonia once per day- he's doing it constantly. You can't drink that much water and not pee like a racehorse
. Also, it's not separating 1/4 clean water, 1/2 clean water, 3/4th clean water, fully clean water, or fully dirty water. It all cycles together. I probably should have just added the amonia, ran the filter and tested throughout the day, but I was trying to see how much water that filter would actually clean in 1/4th a day.
So, does the filter help? Yes. No arguments there. The filter did clean the water, it just took awhile. If it was dealing with a larger tank with more clean water then it could probably do a good job extending the time allowed between water changes, but it's only got 2.5gal to work with.
However, for a 1gal tank you're better off just cleaning the tank daily. Keep the filter though. You might be able to find a cheap 3gal critter keeper and could use it with that. http://www.youtube.com/watch?v=tVD8OysZIbQ
Be mindful of heaters and small tanks though, the water can get too got if it doesn't have a gauge. I use a 25w Top Fin heater (just be sure to unplug it and let it sit for a few minutes before you remove it, and that it CAN burn you if you touch it.
Other than that, be sure you keep that plastic betta cup. Just scoop him up onto it for water changes- nets can hurt them.