How do you beat the island with the museum on Poptropica?
The island that includes a museum in the quest is Counterfeit island.
(see the related link)
There are also museums on Early Poptropica, Shark Tooth Island, Nabooti Island, AStro Knights island, Mythology Island, Steamworks island, Cryptids Island, and Game Show island.
On Counterfeit Island, the curator of the Museum Fantastique summons you to meet her on Early Poptropica (an extraordinary inclusion for Poptropica). Take the blimp to the other island, and enter the Pop Art Museum there. After meeting the curator, return to Counterfeit Island to continue the quest.