In CPH I stayed at the Hilton which is connected to the terminal via a tunnel. It was a nice hotel and had (obviously) easy access to the airport and just as easy access to the metro which gets you downtown in minutes on a very modern train. As the Hilton is an airport hotel, they seem quite accustomed to odd check-in / check-out times.
By the train station itself I noticed an
SAS Radisson Blu. I didn't go inside but it seemed OK from the outside.