Here is the website for Copenhagen airport:
http://www.copenhagenairport.org/getting-to-copenhagen.shtml

Here is another website: http://www.holidaytaxis.com/transfers/denmark/copenhagen-airport/

Be aware that Copenhagen is very expensive! We flew into Copenhagen last year to board our ship.

A couple of other suggestions for you:
1) contact the hotel where you will be staying and ask them if they book taxis and get a price from them. Sometimes it may be cheaper than booking the taxi directly. Sometimes it's more expensive.

2) Have you booked your hotel already? Perhaps see if any of the hotels offer free shuttles from airport to hotel. I don't think city is far from the airport or port so you may find this cheaper for you.

3) Hotel may also offer shuttles to the port.

Have you checked out the www.cruisecritic.com boards? You may find others willing to share a ride w/ you. Look for your Roll Call by cruise line and date of cruise. Hope this helps.

Thank you so much for your response
I had seen the Copenhagen airport site, but the destination points seem to be other cities
I had also seen the holiday taxis but was concerned the pricing is in British pounds
Was hoping to find something more local but may give them a second look, so thank you
Our hotel does not provide transportation
I will check the cruise critic board, great suggestion

We took cab from airport to hotel and hotel to crusie port.
Hotel is Scandic Weber (downtown by Tivoli) Cab fare about 350 Krones one way. There were 4 of us and fit into cab with luggage no problem.

Don't recall seeing shuttle. Many airport hotel offer shuttle to airport and back. Princess offer shuttle from Airport to cruise port. Do the math and see if taxi is cheaper once you group 3 or 4 peopel into a cab.

You are correct, pricing seems to be in pounds, maybe it's a British site.
Guess you could always contact them and ask them what currency payment would be made in.

Good luck.