Hi reregood, thanks for your question! King Street Station (where Amtrak stops in Seattle) is 1.5 miles from our hotel. It's a very slight incline walk, and would take about 35 minutes. Depending on your luggage, time of day, weather, etc., that may or may not be pleasant for you (but I've made the walk before myself, and it's not bad at all).
As other travelers have said, cabs or an Uber/Lyft would be an easy catch. Many know the train schedule and wait outside the station. Depending on traffic, it's about a 10-15 minute drive, so the cost should be around $12-15. We're also on a major bus route, and could provide route and cost information to you, if you'd like to know it.
Hope this helps as you're planning!
Assistant General Manager
Amtrak comes in King Street Station near the stadium/Chinatown International District. There's a light rail that goes from International district Station to Downtown Seattle, Westlake stop. From Downtown you have a bit of a walk to Belltown Inn, but I've done it many times pulling a carry on bag with a day pack on my shoulders. Uber would be another option from downtown to Belltown Which has good bus connections to other parts of Seattle.
Get an answer from Belltown Inn staff and past guests.