The SL pass is actually good for the SL commuter train from the airport. However, with that pass you will have to pay the SEK 120 fee to go through the barrier at Arlanda Central Station.

If you really want to get cheap transportation, a solution could be to take SL bus 583 from Arlanda to Märsta, and catch an SL commuter train from there to Stockholm C. The 72 hour pass is valid, and there's no extra charge if you take the bus to Märsta first.

GUWonder recommends the tourist office at Arlanda. The airport's information desks double as tourist offices, and they sell all kinds of tickets. If you arrive in terminal 5, you can find the information desk 30 meters to the right after customs. If you arrive in terminal 2, you can find the info desk 10 meters straight ahead after passing through customs.