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A three zone ticket will take you from anywhere in Stockholm all the way to Arlanda including any bus or other railbound transport you want to use.
If you arrive at Arlanda by SL train you also need to pay 120 SEK (β¬12) for entry to the airport (free for under 18s). You can either do this in any Access automat, at the train station before starting your journey or when leaving the train station at Arlanda. At Arlanda you may have to wait for one or two to do this before it is your turn.
If you are on a low budget there is also the option to take the train to MΓ€rsta instead and get a number 583 bus from there. This adds about 15 minutes to the journey time. For some reason it is free to access the airport in this way!