Anyhoo, they can live off of scraps and such, but since you've got a betta in there and there's no chance of overfeeding, you could break off small pieces of algae or shrimp pellets and put them in.
Ghost shrimp are really good at smelling out food, and they eat quite a lot. You could also feed them frozen foods and flake foods, but there's a high chance of polluting the tank.
I'm not so sure about the filter, though. Seeing as how all tanks need a filter, you should have that in your tank. As for air pumps and such, it wouldn't hurt to get water circulating and breaking the surface.