There are no bars or cafes on Gorgona. The island is a national park with a single restaurant and no nightlife of any kind. The park store near the main dock carries basic drinks — water, soft drinks, possibly beer — but it is a supply point, not a social one.
Bring whatever you want to drink in the evenings. There is no rule against having a beer on your cabin steps while the jungle makes its noises in the dark, which is a perfectly good substitute for a bar.