Airtstone, broken filter and my fantail?

If you're talking about a fantail betta, definitely. My brother had one in something smaller than a plastic cup for around 5 years with no filter or bubbler and algae resting on the top. lol how sad.. And if you're talking about a fantail goldfish, they're extremely durable and it'll be fine.. especially since your bubble stone is creating movement. Don't worry about it. I used to have one of those in just a bowl in my inexperienced days as well and it lasted quite a while too. lol (: