Good luck with your new Betta! She looks like she needs lots of time and patience, but she's adorable ;0)... she has clamped fins (it could just be stress from a new home, but watch for other signs of illness) and her anal fin is definitely damaged. I would recommend Indian Almond Leaf and/or Bettafix.

Be sure to change her water 2x more than you normally would, to help her heal. If it is less than a gallon, it needs to be changed every 2-3 days! I think it's fine to keep Bettas in 1/2 gallons temporarily, as long as water changes are very frequent, but I would not keep an adult Betta in one forever. 2.5-5 is ideal. You could've bought a 5 gallon clear tupperware for less than that Critter Keeper, for temporarily