I got back from a great cruise on Norwegian Sky on May 31; it had also been my first visit to Seattle. I took Gray Line's Airport Express shuttle bus, which is $8.50 one-way or $14 r.t., runs every half-hour, and stops at eight or so of the downtown hotels. I took it to the last one before it heads back to the airport, the Warwick Hotel (I wasn't actually staying there, but it was walking distance to where I was staying.) On the way home, the Gray Line buses come right to the piers, so you can take advantage of the round trip fare. They have a web site, and can also be reached at 206-626-6088, or 800-426-7532 for recorded schedule info.
A six-by-twenty-block area of downtown features bus service at no charge from 6am to 7pm, which comes in very handy. Ask the Seattle Visitors' Bureau to include a bus map if you request a packet from them; you'll also get coupons good for discounts at local attractions and a lodging guide.
Good eats: right off the lobby of the abovementioned Warwick Hotel (at 4th & Lenora, 206-777-1990) is the Brasserie Margaux. Takes the edge off jet lag, lemme tell ya, and despite having had little to eat all day but airline pretzels, I still couldn't finish their braised lamb shank, which for me is really saying something. I also enjoyed the Asian Fusion cuisine at Roy's at the Westin Hotel (206-256-7697, 1900 5th ave, south of Virginia) and the pan-asian menu at Wild Ginger (206-623-4450, at 3rd ave. & Union) as well as yummy gelato at a place right across 3rd.
Cheap sleeps: The northeast corner of that free bus zone is at sixth and Battery st.; within a couple of blocks of that spot are several lower-cost lodgings, including the Best Western Loyal Inn, Sixth Avenue Inn, Travelodge City Center, Days Inn, and La Quinta Inn. I wound up at that Best Western because of a special offer linked to the use of my bank credit card, but many of them offer goodies like included continental breakfast, free local calls, exercise room, onsite parking, etc. Call around and see what works best for you. The neighborhood is bland, but it's easy walking distance to both downtown to the south, and north to Seattle Center, the campuslike area that features the Space Needle, Experience Music Project, Science Museum, and other attractions.