Sailed on the 7 Night Northern Europe Last August from Copenhagen. I would encourage you to just take a Taxi on Embark day. They can be hailed easily via app and are fairly inexpensive. The app lets you decide what vehicle size you need to call which is an awesome feature. (Van, Car, SUV, etc) Its all a pre-calculated system too so no haggling or anything like that.
When you are Disembarking, its easy to take a taxi from the port. There was a bit of a line when we got off the ship (We weren't in a hurry) but it went quickly.
Because the airport is so far away, We elected to take the train from the airport to Copenhagen Central Station which saves a good chunk of change. Its a good length drive otherwise (30 ish minutes minimum) which will be much more expensive in a taxi. We had the Taxi driver drop us off at the Train station instead of driving all the way back to the airport as the cost would have been astronomical compared to taking the train.
I highly encourage you to download the app "Rome2Rio". The app allows you to enter a location and will show you all the ways you can get there. (Taxi, Train, etc.) it is very helpful for international travel because it presents additional options you may not normally look for. I did a lot of research on the app in advance to see what the best options were.
Have a great sailing! We look forward to sailing out of Copenhagen again, its such an incredible place.