I'm not sure about this, but given the distance from Seattle-Tacoma International Airport to Downtown Seattle, I'm not sure if there are courtesy shuttles.
As far as geographic location that you're looking for, the Quality Inn and Suites - Seattle Center, and the Comfort Suites Downtown / Seattle Center, are within walking distance to the Seattle Center Monorail and Space Needle.
Also, you may want to consider posting this in the "Embarkation ports, hotels, and transfers" thread.
2007 - Majesty of the Seas
2009 - Enchantment of the Seas
2010 - Freedom of the Seas