These are useful tips, for arriving in Sweden via Arlanda Airport and using public transport to get to Stockholm City. Ultimately, the smartest way to get around Stockholm City is to you get a transferable SL card. An empty card costs 20 Crowns (Krona) = 2.47 USD, then you can recharge charge your card with the relevant credit for your stay, be it 24 hours or 1 month.
