Hi!
You can certainly take a taxi, which would the most convenient. I don't know the cost, but the airport is not that far from downtown. 20 minutes or so. A bit more in rush hour traffic. There is also public transportation, and there might be some express buses. You could also catch the train that runs every 20 minutes between Malmö and Copenhagen, and stops at Copenhagen Airport, and then catch a taxi from the train station. Not knowing how many in your party I don't know if this would make financial sense.
Plenty of hotels - all depend on what you want to pay. Radisson SAS Royal Hotel is one of the top hotels (at Hammerichsgade 1 and right across the street from Tivoli Amusement Park and pretty much walking distance from anyplace in central Copenhagen). Another good hotel is Radisson SAS Scandinavia Hotel at Amager Boulevard 70. Maybe a mile or so away from the central area but there are frequent buses. Obviously plenty of others. You could just google for 'hotels in Copenhagen'. If you want to know about a particular area, let me know. I have sent you a message thru the Forum with some additional information.
The cruiseships dock at Langelinie and the Free Port, both of which are a couple of miles north of the downtown area. The Free Port a bit further away. And as far as I know, no hotels up in that area, but there are plenty of hotels in the central part of Copenhagen.
Aloha!
Bo-Göran