The most reliable way to get to/from ARN is via Arlanda Express. 20 minutes each way. It is a bit pricey.

If you are traveling this summer, and assuming your flight arrives early in the morning, I would recommend getting to the Vasa Museum as early as possible (they open at 8:30am through August 31). Since you're an outdoorsy person, perhaps a stroll through Skansen (a nice park/zoo with good views of the harbor) nearby then hitting the Vasa Museum a little later in the day to avoid the initial crush of tourists. After the Museum, you can take a ferry across the harbor to Gamla Stan and walk around. It's about a 10-minute walk from the north side of Gamla Stan back to the Central Station to catch the Arlanda Express back to ARN.

If you're traveling in the fall/winter, then the crowds won't be as bad at the Vasa Museum, but it doesn't open until 10am, so maybe you should still hit Skansen first.

I typically have four hours between my flight landing and taking a train to my final destination. That still gives me plenty of time to walk around Gamla Stan and eat at a random café before walking back to the Central Station.