I am Danish and live in Copenhagen, so just my two cents.
You can easily take public transportation and/or a taxi, it is completely safe. Also alone. Alot of people are using public transportation, especially in the city centre, so you will not even feel alone. Most of bus drivers are very helpful if you're not sure where to go.
Taxis are very expensive in Denmark, and when we've cruised from Copenhagen we've taken a bus all the way to the terminal.
I suggest taking the metro to Orientkaj (last stop on the line) from CPH Central Station or from Kongens Nytorv (which even connects to/from the airport). From Orientkaj you can walk (approx. 2 km.) or take a bus to the terminal (Oceankaj). Bus 25 and 164 both operates this route, and bus 25 ONLY operates this route and drives when there's a ship in port.