The Thomson Spirit docks in Kalundborg, or at least it did two years ago when we were there. It is about 45 mins from there to Copenhagen. We went on the Thomson organised transfer as we were guaranteed not to miss the ship when we returned rather than risk local taxis given the distance. Think the transfer was about £20 each. There did not appear to be any locally run shuttles at the port.....but someone may know differently.
A number of cruise ships dock at Langelinie pier, a short walk away from the Little Mermaid statue. If you enjoy walking you could walk along the water front, about 30-40 mins to the canal area of Nyhavn, passing the Amalienborg Palace on the way. A canal trip will give you a good overview of many of the city sights from the water. If you are not able to walk long distances, the No.26 bus stops at Langelinie pier, this will take you into the main square in Copenhagen close to Stroget, the main shopping street. I believe the HOHO bus also stops at Langelinie. Opposite the cruise port there are a few tourist shops and the Copenhagen Information Centre where the staff are very helpful and speak good English, as do most Danish people. Taxis are plentiful from the port and reasonably priced. I have never cruised with Thomson, so I cannot confirm whether they dock at Langelinie. The other dock for cruise ships is the Freeport terminal. I'm sure other contributors will be able to give you more information. Happy cruising- Jan.
I just checked the Thomson site and it appears that the Spirit does not dock at Copenhagen at all this year!. They use Kalundborg which is some distance away. I also checked the Port of Copenhagen site for cruise ship arrivals and lots of ships are listed but Spirit is not one of them.