It is possible to cycle a tank of any size; the smaller, the more difficult. Due to the size, ammonia reaches toxic levels before bacteria can establish itself. In larger tanks, bacteria starts setting up even when the ammonia levels are still at safe levels.
A lot of people (myself included) generally tell beginners they are unable to cycle a smaller tank, due to the difficulty and upkeep levels required to establish and maintain it.
Larger tanks are much easier to care for.
And yes, I recommend changing the water everyday until he gets better. Make sure the temperature is consistent and you're allowing him time to acclimate him to the new water in a floating cup before releasing him into his squeaky clean tank.