Should a Betta fish be going up to the top of the fish bowl to get air?

Yes, bettas should go to the surface for air. They are air breathers. They have an organ, the lybrinth, that is very lung like. They breath in air from the surface. Their gills do not work like other fish. If they can not get to the surface for air, they die. Of corse yes their gills are different from other fish so that is why a Betta fish has to be going up to the surface to get air or else they die.