Qantaslink operate daily flights from Cairns to Horn Island. There is no airstrip on Thursday Island therefore one lands on nearby Horn Island. A ferry or water taxi transports guests then to the island.
We took a short tour of Thursday Island with Peddell's Thursday Island Tours in what I could imagine is the only tour operator in this part of the world. The guide Sue was great and she gave us a comprehensive overview of the history and other aspects of the island.
Seaswift operate a cargo barge with passenger berths weekly from Cairns to Thursday Island and Seisia. They can ship you and your 4WD back to Cairns if you only wish to drive one way to Cape York.