Can i use an air pump instead of a filter to keep my 10 gal bacteria alive?
Not an air pump by itself, but you can use an air-driven filter (together with an air pump) instead of a power filter. The current from the air-powered filter would probably not be as strong. I recommend the sponge-type filters like (for example) ATI’s Hydro-Sponge for use with air pumps. Bettas, as you have noticed, are not very strong swimmers (especially the long-finned types). They tend to live in fairly still waters, sometimes even stagnant water, in their homelands.