My husband and I will be sailing on the Star for the first time in October so I can't tell you anything about the ship. We have done the Western itinerary several times so I can help you there.
In Grand Cayman, you can go to Singray City and snorkel with the stingrays. You can book either on board or get a cheaper rate by booking ahead of time. Some companies are NativeWay Water Sports (we have used them twice. I booked by e-mail and paid at the pier with no advanced payment); I've heard that Captain Marvin's is also good.
You can go to Hell and the Turtle Farm also. I can't recommend any companies for that tour.
In Cozumel you can go to Paradise Beach. It's a free beach, but you pay for the water sports. They give you a chair and there is a restaurant there. You can go to Chanknaab Park. It's not free, but doesn't cost much to get in (sorry, I can't remember how much). You can spend the day there either on the beach or you can book a dolphin swim there.
You can also rent a jeep and drive around the island. On the "other side" of the island it's still pretty much untouched. There's a bar at the end called Mezcalita's which is very relaxing.
If the ship goes to Ocho Rios you can climb Duns River Falls and take a tour through Fern Gulley. We booked ahead of time through A-Z Planners. I did everything by e-mail with Belinda and again paid when we got there. If the ship goes to MOBAY, I can't help you -- never been there.
E-mail address for Belinda at A-Z is firstname.lastname@example.org
The website is: www.a-zjamaicaplanners.com
For NativeWay, contact Sharon Ebanks at nativway@CandW.ky
I don't have the website address readily available, but you can do a search on Native Way and find it.
Hope this helps.