I wasn't willing to shell out 700 Swedish crowns or so for a cab ride to the airport, so I decided to take the train instead for the low low price of 100 crowns.
The brother was so skeptical that he offered to give me cab fare, but I persevered.
However, I did get him to help me with my luggage up to the station. Later I had to change trains, but the other train was just on the other side of the same platform.
Got to the airport without a hitch. It was quite a long walk to Terminal 2 though.