Personally, I wouldn't bother filtering something this small.
This is what I wrote on another person's thread about the same thing..
I wouldn't filter anything smaller than a 5g because it just creates current the fish can't get away from it. Also trying to cycle anything smaller will require more careful monitoring and more supplies and it probably still won't ever fully establish.
If you did want to do that I would use a sponge filter, with pump and airline. Then you will need a drops kit for ammonia, nitrite and nitrate. You need to be testing daily with a reliable drops kit for ammonia and nitrite and doing an extra 50% change any time you see either. In addition to this a weekly 50% with siphon or new fresh turkey baster that has never seen chems is needed to remove poop and other debris from the gravel. It is not enough to just scoop water off the top ever.
First you will see ammonia, then nitrite. Eventually, hopefully, you will see ammonia fall and stay at 0 even after a week of no water changes, and finally nitrite. At this point you will be left with only nitrates after a full week of no changes and these can be kept <20ppm by twice weekly 50% change with baster/siphon. However, cycling will take up to two months to complete and many if not most people will tell you that you can't cycle a tank of this size and you will always need twice weekly 50% and 100% water changes or you will always see ammonia continuing to build and the cycle will never finish..
If you don't cycle it and while you are not cycling it you should do two water changes a week of 50% and 100%.
The 50% changes the betta can be kept in the bowl and use a turkey baster to remove half the water and as much of the debris as possible. For the 100% you need to remove him - scoop him out with a plastic solo type cup and set aside while you thoroughly rinse the bowl and gravel to remove the debris. Then he should be acclimated to the new water by floating for an hour while you slowly add a couple tablespoons of new water to the cup every 10 minutes. When you release him, try to let as little of the old cup water back into the tank as possible. All water changes should use same temp water, matched to running tap using the in tank thermometer and the water needs to be premixed with conditioner before adding it to the betta tank. If you don't already have anything, you can use gallon water jugs from the grocery store - rinsed thoroughly in hot water but no chems.