I think you can probably find one for less than $800, but it will be rented within 24 hours or so after it is listed for rent. I checked Belltown and Capitol Hill, both near downtown, and over by the Space Needle at the base of Queen Anne seemed to be the best value. On the Eastside, prices were not much different. Seems to me prices have gone up quite a bit in the last 18 months or so.
Many one bedrooms that have not been rented quickly are in the $1,100 to $1,200 range.