Can I use my Busch gardens cups and popcorn buckets at adventure islan?

Yes

Yes, certain cups are refillable. If it was a "99 cent all day refills" cup, purchased for the 2019 season, you can use it at all four Florida amusement parks (Busch Gardens, Adventure Island, Sea World, & Aquatica). I'm unsure about the popcorn, but if it was advertised as a "discounted refills" bucket, it's plausible you can use it at the other parks as well (just like the cup).