Most people who get betta fish are told that all you have to do is plop them in a small bowl and change the water every now and then, and the fish will live for maybe a year just because they are hardy fish - but this is NOT ideal. Think of it this way - you could survive for a while in a small cold closet with no sanitation system, but it's much better to have a long life in a house with a heater and a toilet.
I would recommend upgrading your set up...bettas in unheated and unfiltered bowls are prone to disease and will probably get sick eventually - resulting in expensive medication or death of the fish.
I'd use a larger tank (2.5 gallons or more) with a heater and filter. Tap water is fine as long as you use water conditioner to get rid of chlorine and other chemicals which can poison a fish.
You should also read the sticky post at the top of this forum for more info on betta care and setting up your new tank. If you follow their advice, you'll have a happy, healthy fish for several years instead of a few months!
Until you can purchase the new set up, make sure that you leave the replacement water out for 24 hours, this way the harmful chemicals will evaporate and don't change it out all at once - do 50% water changes every week with water that is about the same temperature as the water you are changing out, so as not to shock the fish.