Where do you find the offering for Poseidon on Mythology island?

Poseidon's realm is the far right side of the island, past the Temple of Muses and the realm of Hades. You will meet Aphrodite and take a test on gods and goddesses. You will also acquire the Giant Pearl and the Hydra's scale.

Use the starfish (from Poseidon's statue in the museum) as an offering on the altar.

To get to the throne room (except as a glitch), you will need Hercules. Hercules will not help you until you have returned all of the 5 sacred items to the Tree of Immortality. Get the Touchscreen Mirror from Aphrodite on Poseidon's beach, so that you can transport Hercules to the realms of Poseidon and Hades (he does not like to walk).