Take the Sound Transit light rail from Sea Tac airport to downtown Seattle. Then take a Metro bus, the D Line, to the Ballard neighborhood (15th Ave W and Market St. stop). Swedish Ballard is 3 blocks west. That should be under 2 hours and under $10 or so.
Ty Irina I'm coming from north Carolina
Yes, what Jim suggested - take a train from the airport to the University of Washington station. Then you will cross the street (use an elevator and the bridge over the road) and take a bus #44 to the Swedish Medical Center (Ballard). It will cost you around $5 max $10.
You could take the 44 bus to the uw light rail station and then the train to the airport
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