Copenhagen is one of the easiest cities to get around. As soon as you exit from the airport, taxis are waiting for you and then just tell him you want to go to the Port.
The airport is about 15 minutes from the city and the port is very close so the whole journey shouldn't take more than 20-25 minutes.
Most people speak English so you shouldn't have any trouble being understood.
The taxis take Danish Kroners or credit cards. I think there is a charge for the cards. They will not take foreign currency and that includes U.S. cash.
Jennie.