Make your own way to the Positano pier in the morning for pickup by your captain.
Step aboard your private vessel and head north along the Mediterranean coast toward the island of Capri. Stop first at Li Galli Islands, a private archipelago and marine reserve. Stop for a swim, or simply relax and enjoy the island landscape, which includes three private luxury villas high above the rocks. These islands are understood as the place where Ulysses hears the song of the sirens in Homer's Greek classic The Odyssey; the islands were also once owned by the Rudolf Nureyev, the great Soviet ballet dancer.
Next up, pass the active volcano Mt. Vesuvius as you continue across the Bay of Naples to Capri. Upon arrival, your captain shows you top sights such as the Green and White grottos, Blue Grotto, and the Faraglioni rock formations. Have time to take a dip in hidden coves and inlets.
If you get hungry, your captain can pull up at the dock of one of Capri's fine waterside restaurants so that you can enjoy a meal or refreshments (own expense).
Spend your time on the island at your leisure. At the end of four hours, your captain returns you to Positano to complete your tour.