At the hotel, you check in normally for your stay, then you check in at a separate desk for the cruise. You can do this anytime during your hotel stay when the cruise desk is open. Basically they do exactly what the port check-in folks do.
The hotel portion is settled up like a typical hotel stay. You have to put down a credit card when you check in and all of your room charges are settled at the end of the stay.
For the sea portion, you have to fill out one of the cards in your documents that specifies what credit card you want to charge. This document, along with several others, is handed in at the cruise desk. Yes, it can be the same one as the card you used for the hotel, but they don't automatically transfer the info for the sea portion. The cruise is then settled at the end as well.