We used the route described above, metro to Norreport, then changed to an S-train to Nordhavn. The description given above by the RS staff sounds like it's in the same general area as where most of the cruise ships are docked.
The metro gets you into town faster from the airport than any other option if you're going to the area at the north end of the Stroget or further north (roughly 15 minutes to the Kongens Nytorv stop, which is adjacent to Nyhavn). The train is a better option if you want to be near City Hall/Tivoli Park area (about a 25 minute ride).
I can't remember the name of the little (by U.S. standards, normal for Europe) store we used most often but it was walking distance from Kongens Nytorv. Ask someone local, they'll know. As I recall Kongens Nytorv is one stop prior to Norreport on the metro line coming from the airport.
The store sells wine at EU prices, so a bottle of French wine is pretty cheap.
Your cruise will let you bring a bottle of wine each onto the ship at each port. It's a good way to save, and drink better wine, versus what's offered on the ship.
We carry one bag where ever we go, including for a cruise - which makes public transportation options more convenient.