Honestly, a store bought sponge filter would be better than a home made one and would work more effectively and efficiently. Sponge filters the size required for a 2ish gallon tank are four dollars and under. Isn't it worth four dollars to make sure your fish has a superior end product? I think it is.
Please use a store bought sponge filter and air pump. That's my recommendation.
Honestly though, the more I think about it, I think a nano HOB
would be better. It will disturb the surface of the water less than the sponge filter, allowing the betta to make a bubble nest. (a bit of hornwort floating in the tank would help him do this)
You could use one of these filters:
Or one of these
All three options (sponge, internal, HOB
) are viable. All 3 (incl air pump for sponge filter option) will cost you $15-20 all-in.
Don't forget to cycle your tank (using the fishless method where you add ammonia, which is the most humane method) before you put fish in.