Well ,the Arlanda Express goes between the Airport and Stockholm City.
However, the area right around CityTerminalen (where the Arlanda Express terminates) is not very nice.
Gamla Stan (the historical town) is about 20 minutes away by walking.
Make sure, you grab a map of Stockholm, otherwise you would be pretty disappointed.
Avoid Slussen and CityTerminalen, when you're in Stockholm - these places are dirty and pretty filthy.