The Beihai Hotel (Qingdao Beihai Binguan) certainly has one of the best locations on the mountain, with quick access (a hop, skip, and plenty of steps) to spectacular sunrises and sunsets. It was built in 1958 but has been renovated and expanded over the years.
The food is spendy, but ingredients are humped up the mountain by man-power, so think of the value of that when you start to complain about the prices. The best cup of coffee on the mountain is in the lobby and the hotdog vendor right outside is a nice option for the economical traveler.
Heat can be inadequate in the colder months, but quilted jackets and extra blankets are provided. Don't expect the service or the facilities to be as top-notch as the scenery, but be glad that you'll only have to wake up five minutes before sunrise to throw on your complimentary (loaned) jacket and head out to where the real greatness is.