Here is the most useful information for the 2033 opseason.
Pier Information
For pier location use this page, its updated closer to arrival with exact pier number and terminal.
http://www.cmport.com/ships-in-port/cruise-ships/season-current
C177 Nordre Toldbod
https://goo.gl/maps/AoV7G8HbFj32
C190-199 Langeliniekaj
https://goo.gl/maps/XgJyRRmopbB2
C331-333 Oceankaj
https://goo.gl/maps/3EnfQ4Za7bN2
Transport to the piers
Nordre Toldbod is better reached by Taxi but bus 1A stops at Esplanaden (Grønningen) when travelling from downtown and when going to downtown its the bus stop called Esplanaden (Store Kongensgade)
Langeliniekaj is served by bus 26 from the Indiakaj bus stop at the bottom of the pier
Oceankaj is served by both bus 25 and 27 stopping at the Oceankaj bus stop next to Terminal 1, they are both running from Østerport station (25 also from Nørreport)
look at www.journeyplanner.dk for schedules.
Single ticket (2 zones) to the downtown area is 24 kroner,
Single ticket to the airport (3 zones) is 36 Kroner.
Or if you want a citypass ticket you can order it on your phone and have it ready when leaving the ship.
Citypass is available in a small version covering Copenhagen and the airport and a large one that covers the entire capital region(so both Kronborg, Frederiksborg, viking ship museum etc)
https://dinoffentligetransport.dk/citypass
Taxi companies
Dantaxi
https://dantaxi4x48.dk/english/
Price calculator
https://dantaxi4x48.dk/hvad-koster-en-taxa/
4x47
https://4x27.dk/english-taxi
Private car
http://cphlimo.com/en/home
Book your own bus if you are a big group.
https://www.bookingabus.com/en
And at last, a useful link if you don´t want to handle your luggage to and from the airport you can let these guys do it.
Its very useful if you have a late flight out of Copenhagen and wants to do some sightseeing before heading home.
https://www.valizo.com/
A big disclaimer
Metro line M4 opens in the spring, I´ll expect that it will cause changes to the busses out at Orientkaj, But I will post when anything is announced.