jossie, I keep my 50L tank heated. the heater is set to 22 so slightly on the cooler side of a tropical tank, but they can be kept in unheated tanks. off topic here, but try adding some fast growing plants and a light. you should see him perk up abit. they are naturally slow moving because the flowing fins prevent them from moving about fast. fyi, bettas always look unhappy. I guess it's the fins.
have you started looking for a tank? no harm in starting early, it gives you time to prepare, especially if it's a used tank as they often need cleaning and sanitizing. mind you there are often some fantastic ones out there going for cheap.
just throughly check the condition of the seals, glass and the cross beam at the top before buying it, avoid ones that are obviously cracked, and be sure the cross beam is secure if the tank is fitted with one. run your fingers over the sillicone on the inside of the tank and check to see that it's not broken or peeling away.
also, ask what side of the tank was used as the front, so you can make this side the at the back when you set it up, I find that the back panel of used tanks tends to have less scratches than the side used for the front by the previous owner, due to the fact that the back was probably never exposed to anything other than a wall.