Bettas have what's called a labyrinth organ. It's like our lungs, they can breath air from the surface. In the wild, there's hardly any aeration in the water so this organ was developed. They rarely take oxygen from the water so yeah. You don't have to use money on an aerator for them! xD
They use their gills less in the cup because the pet stores do not change the water often enough and it is usually full of ammonia. It does not have anything to do with the amount of air in the water. Bettas do not need air pumps. I have one but thats because the filter I use is air pump driven.