There's no lodging in the immediate area of the pier. The ship departs from Smith's Cove (Pier 92). That's outside of the downtown area (the pier downtown is the Bell St. Pier - Disney's not using it).
You can stay nearly anywhere and arrange shuttle transportation to/from the pier for the same price. Shuttle Express
is just one of several companies offering the service. Hotels in the area of Seattle Center are about the closest to the pier, although anyplace downtown is no more than a mile or so farther than that. There are plenty of decent possibilities. At this stage, I'd just shop Kayak.com or Expedia.com and see what's available. Setting your sights on one or a handful of spots could cost you. There are two hotels in the South Lake Union area - they're about the same distance from the cruise pier, and still very convenient to downtown. Meantime, the rates tend to be a bit lower for the value delivered.
If you're planning to do some in-town sightseeing, I suggest you do stay in town. If you just need a place to rest your head before the cruise, there are plenty of decent, reasonably-priced places near the airport. The shuttle services serve those as well.
Here are hotels I've personally stayed at:
Executive Hotel Pacific - rates here tend to be low for downtown. It's clean, basic lodging, in an older building
Warwick Hotel - Closer to the cruise pier - it's a 60s-vintage hotel that shows its age a bit, but still clean, comfy, and well-run
Hotel 1000 - If you have the bucks - boutique-style high rise. Large, well-appointed rooms, amazing bathrooms. Ranks very highly in guest surveys, and I have to agree.
Hotel Monaco - Across the street from Executive Pacific, much nicer. A Kimpton hotel, so near the higher end of the price range.
Belltown Inn - Can be bargain-priced, but also very quirky. Very basic furnishing, no air conditioning (but it's not essential in Seattle). Otherwise, clean and well run.