I kept bettas in a 1.5 before, and while it's not ideal, there are several things that apply to tanks of all sizes that can improve his life. Really, a 5 gallon or even a 10 would be best, but I understand the money thing. Check craigslist, the classifieds, garage sales, thrift stores, and so on. You can get amazing deals on tanks. (I've seen a 6' long tank at GoodWill for $14)
I cannot stress this enough, live plants make a huge impact on fish and water quality. You may have to replace the stock lighting that comes with a tank, but that can be as easy as a desk lamp and the right kind of light bulb. Many plants like mosses, anubias, crypts, and swords are very easy to grow, and many can survive just about anything you can throw at them. They will help absorb ammonia, nitrites and nitrates, and keep the water healthier for fish.
Water changes and cleaning
another important thing, as important as the plants. This is the number one way you are going to keep your fish happy. Small tanks are very difficult to cycle (don't know about cycling? read up The Fishless Cycle
) since everything can go wrong so quickly with them. For the first month or two, I suggest bi-weekly 20% water changes. Make sure to dechlorinate the water and have it the same temp as the tank water. Gravel vacuums and water pythons can make water changes easy in larger tanks, but might be too large for a small tank. Try using airline tubing instead to siphon water and clean the substrate.
make sure the water temperature stays consistent. Bettas can handle warm water, due to their labyrinth organ, but cooler temps can be quite bad for them. Heater or no heater, keep in mind that in a tank that small, temps can and will change quickly.
A high quality betta
food will allow the betta
to process his food better, so there's less waste at the end. Never over feed, since this creates waste and ammonias. At lower temperatures, betta's can't digest food, so this can be dangerous. Try mashed up peas (just tiny bits), to help get things moving, and feeding less.
I don't know much about the filter that comes with that kit. It might be too strong, so you'll want to tweak it a bit if it looks like betta can't handle the flow.
That's all I can think of for now, hope this helps.