In that small of a tank, an air-stone would probably be more of a detriment than benefit unless it was *very* low power---too many bubbles in a small space leaves nowhere for the betta to get way from them and rest. And, after all, labyrinth fish--able to take air from the surface, so air exchange not as big an issue as it might be for a similar bioload with a different fish.
Biggest thing is keeping a good temp and pristine water conditions with regular water changes. Easiest method is really to get two containers and keep a supply of treated,aged water on hand to do an easy daily 100% change.
Rinse empty container with water, transfer over any hardscape/plant material, fill part way, scoop (or pour-some vases lend themselves to an easy transfer, some don't) betta in, fill the rest of the way. Dump and rinse the previous vase. Top off whatever water storage container(s) you're using so you'll be ready with treated, aged and room-temperature water for the next change.
+1 on providing at least some plant material. Even a freshly trimmed pothos (terrestial plant) or amazon sword leave floated in the bowl will give the betta some sense of shelter and something to explore.