Before you do anything else, change at least half of the water with fresh that has been treated. Depending on where you live, your betta may be cold and is almost certainly being exposed to ammonia from his own wastes. The water change will help with the ammonia.
Bettas need their water at least 76F and the temperature should be stable. In my house, an unheated tank would likely be at 70F or less this time of year. You should get a heater to keep the temperature stable and high enough. For such a small tank, a 25W heater should be plenty.
Change at least half the water every other day and read up on cycling the tank. By changing half the water every second day you should be able to keep your betta healthy until you get caught up on the information you will need to do the right testing and get the gear to do it right. This is not hard but lack of information of their needs has killed many a fish.
To set it up properly for your fish, you will need a liquid water quality testing kit, a thermometer, a heater and I would get a small filter like one made for a 5 gallon tank. To really do it right, give him something to hide behind when he's not feeling secure.