Can a Betta live with having a filter and air pump or just a fish bowl?

Yes, but not for long. Bettas are no different from other fish in their requirements. The basic rules for successfull fishkeeping are as follows. :- 1 inch (2.25cm) of fish needs at least 1 gallon (3.78ltrs) of water. Every tank needs a permanently running cycled filter. :- Every tank needs at least 50% of its water replaced every week. If you follow the above rules, feed the fish the correct diet and keep the container covered and at 78F a Betta will do OK. I can guarantee that if you don't keep the preceeding rules that your fish will have constant health problems and will live a considerably shorter life.