Welcome to the forum! Your boy is very pretty...he's got some very nice fins. He seems like a Pokey to me...
Anyway, don't let the small size deter you. A 2.5+ gallon tank is recommended, but if this is what you have for now your buddy will be okay so long as you're willing to do regular water changes. It looks to me to be about a .5-1 gallon tank...so at the very least, do one water change every other day. Not sure if you know this already or not, but make sure not to wash anything out with soap! Just hot water will do the trick.
For now, I'd also recommend giving him a little more swimming room by filling it up a bit more. You were right to lower the water level, since they can and will jump...but instead, maybe you can find something to place on top of the bowl? Like plastic wrap with a few holes punched in or some mesh.
Also, be sure to get him a heater! A heater is a must for a betta, unless you live somewhere where it's constantly 75-80 degrees.