From Sunday 9th of December 2012 you can go with the commuter train directly to Arlanda airport but you will need a special ticket (75 SEK) to enter the Airport.
Trains will leave with 30-minute traffic from early morning until about 22:00. On weekend mornings and late evenings, they depart every hour. Travel time from central station will be just under 40 minutes.
So, important to remember that you need a special ticket when traveling to or from the airport.
If you don’t already have a SL ticket, there are single tickets including Arlanda Passage at a cost of 125 SEK. About the same price as the Arlanda Airport bus.
The train to catch is the commuter rail 38 heading for Uppsala.
Have a nice trip!