I agree with the concerns about traffic in downtown Seattle during morning rush hour. If you plan on being in or near downtown Seattle between 7 am - 10 am, best be prepared for the congestion and know your options.
Light Rail from SeaTac Airport
A bit of a walk from baggage claim to the light rail station, but very doable if you're up for it and can manage with your luggage in tow.
See the route you'll walk (in blue) -
A video of the walk here (watch the video in the Directions to Link light rail section)-
https://www.portseattle.org/Sea-Tac...ound-Transportation/Pages/Public-Transit.aspx
Get off at Westlake Station in downtown Seattle. There are signs you can follow to get to the exit above ground that you want. I suggest going to the Westlake Center exit to catch a taxi/Uber/Lyft to Pier 91. 3rd Ave is a bus-only zone from 6am - 9am on weekdays, so other vehicles won't be able to pick you up from that side.
Bus Option from Westlake Station
If you really want to save on your fare to the pier, it's just one bus ride from Westlake Station. You can take
Bus Route 24 to W Magnolia or
Bus Route 33 to E Magnolia. Either one will take you to Pier 91. You can catch either bus on 3rd Ave and Pine St. Follow the signs to 3rd Ave and Pine St exit when you get off at Westlake Station from light rail.
Taxi/Uber/Lyft from SeaTac Airport
Take 509/99 route suggested above, but instead of taking the Western Avenue exit from the viaduct, I suggest taking one of the street exits (Thomas St or Harrison St) after the Battery Street Tunnel, then head SOUTH on Dexter Ave N (right turn from Thomas or Harrison), and then head WEST on Denny Way (right turn from Dexter). Denny Way will take you to Western Ave after several stop lights. Western Ave will take you to Elliott Ave W, which will take you to the exit (W Garfield St/Magnolia Bridge) for the pier. Follow the signs to your cruise terminal.
I suggested the alternate route because Western Ave exit is usually very backed up during morning rush hour, which may substantially increase your fare. If not too backed-up, I would take that exit and it's a straight shot to Elliott Ave W from Western Ave.
Hope these help.