Hello,
Been to Cleopatras Island about 3 times, but not on an excursion. I normally drive to the harbour and take the small water taxi out. Takes about 30 minutes to drive to the harbour, 2nd left off of the road from
Marmaris to Govacek and the harbour is down on the right about a mile or so past the Breakfast place (forget the name). I think it was about 15 TL for the water taxi, takes about 20 minutes and there is an entry charge when you get to the island. belive the last water taxi goes back around six.
Very nice island, well worth a visit