Does anyone know the availability of taxis at Oceanjaj or Nordhaven cruise terminal at 6am or so? What should we expect to pay?
The information you're requesting is at http://cruisesfromcopenhagen.com/Resources/AirportTransfers.aspx
You really should have got such a later flight out. Good luck.
Are you really sure you will be able to disembark at 6:00 am? If that is the time the ship docks, it will likely be closer to 7:00 when you walk down the gangplank, even with priority disembarkation.
Will the cruise be arriving the day before and overnight in Copenhagen? If so, you can make arrangements ahead of time with the taxi to pick you up that early. I am also cruising with HAL, but our flight leaves at 10.
There are usually taxis waiting, it shouldn't be hard if you hold your bags and get off the boat first thing in the morning.
We took a Baltic cruise out of Copenhagen (with only carry-on, so getting off wasn't an issue). When we got off the boat, we walked to the train (kind of a long walk out of the port area, but Nordhavn is closer than our Princess cruise in Frihavnen), went a short way to connect with the metro at Norreport, then to our hotel near Kongens Nytorv.
If you did the same and didn't get off at Kongens Nytorv, you would be at the airport in under fifteen minutes from there - faster than a taxi.
The part I'd worry about for you is the walk to the train, especially if you pack heavily. Trains run regularly as does the metro but there will be plenty of commuters unless it's a weekend.