They are tropical fish and need a heater, unless your room temperature is in the upper 70's or lower 80's at all times. Get a thermometer to check his water temp. The ideal range it 76-82. You can get a good heater for $10 at most wal-marts or pet stores. If that is not a possibility, you'll need to put a lamp very close to his tank water during the day, and wrap his tank at night with warm towels to keep the warmth in the tank.
You can use tap water as long as you have a good water conditioner that removes the chlorine. Also, with that size of a tank, you're going to need to 100% water changes 5-6 times a week. A bigger tank is easier to maintain, clean and heat. The recommended minimum tank size is 2.5 gallons (The general rule for all fish being 1 gallon per inch of fish - and since Bettas are generally 2 1/2 inches long, 2.5 gallons is the absolute minimum). Remember, fish naturally produce ammonia in their waste, and ammonia is toxic to fish, so you should also get a water testing kit to make sure that the ammonia levels never rise above .25 (even that's pushing it), and if they are, you'll need to do a water change immediately, or risk killing your fish.