Don t know about a free shuttle, but a Seattle Metro bus (very cheap) will take you right from the airport to the Greyhound Bus Station at 811 Stewart Street.Take the MT (Metro Transit) Bus number 194 to the Convention Center and Tunnel Station A. Then, walk less than 1/4 mile west to the bus station by:Walk a short distance SW on Convention Place Acrd. Turn left on Pedestrian Stai. Walk a short distance SW on a significant grade on Pedestrian Stai. Turn right on 9th Ave. Walk a short distance NW on 9th Ave. Bear left on Olive Way. Walk a short distance SW on Olive Way. Turn right on 8th Ave. Walk approx. 1 block NW on 8th Ave. Total walking is 0.25 miles.
They do have a free shuttle.