I don't think you can leave an island with a horse.
In Jungle Island
On every island, the Poptropica blimp is above Main Street. To leave Nabooti Island, return to Nabooti and walk right from the Museum.
A
you just get in the plane
You can only leave from Dusty Gulch, where you entered. You can also exit using the Haunted House from the Poptropica store, which exits onto the Island Map. If you return to Wild West Island, use the whistle to call your horse.
any planet
To leave, use the Poptropica blimp on Main Street. (To complete the island, see the related question.)
go into the desert with it.
Wait for principal to leave then you leave
Leave the castle, enter the city, and jump up into the blimp.
In Wild West island you put your mouse over the horse and it'll say something like ride horse or dismount horse.
You click on the boat near where you came