I've done this with every single one of my bettas and it's worked fine so far:

-Preferably set up the tank before getting the betta, so that the temperature will get to room temp. If you don't, it's alright though.. sometimes I just get water straight from the tap and raise it to about 73/74
-Add dechlorinator. Bottled water isn't necessary. You can get a small bottle of dechlorinator for a few bucks. I get a big bottle for $12 and it lasts for MONTHS with about 16 bettas
-Float the bag in the water. You want to do that so that you can acclimate him to the different temperature. Also, it's a good idea to slowly add water instead of just dumping him out once the tempature reaches room temp. You also want to get him used to different PH levels and whatever differences his water and your water has.
-I do weekly 100% water changes with my bettas (most are in 1 gallon tanks). With a 2.5 gallon, you might be able to get away with 1 and a half weeks.. but it's hard to do too many water changes.

I do 100% water changes with my tanks that aren't cycled.. anything that is smaller than 5 gallons I believe, is too risky to cycle as it won't be as stable.