I know that for the Med cruise we took last year, the buses were rolling into the port about the same time we arrived by taxi ... around 11 am, if not earlier. Not sure if it will be the same in Copenhagen.
You need to ask
DCL if you don't stay at the DCL hotel, can you still get the transfer from that hotel. That might save you from having to stay at the airport or heading back to the airport. If you're only getting into Copenhagen the day before and have no intention of sightseeing, then you might as well stay at the airport. I suspect it's cheaper there than in the city.