I am going to Nagoya,Japan in May. I am travelling by China Eastern which has a change of planes at Yantai. Stoppage time is 9hr 30 min(0:55am to 10:35am). I need a cheap accommodation with 24hrs checkin to stay close to airport.
The new Yantai Airport, Yantai Penglai International, is really remote and is not even in the city proper of Yantai. If you don't want to go too far, your best bet would be the Yantai Airport International Hotel (烟台机场国际酒店), which is right at the airport . One day's stay costs about 400 RMB, or ~58 USD. You could book the hotel on TripAdvisor or Ctrip.com.
There is also a GreenTree Inn chain hotel around 10 km away from the airport, called the GreenTree Inn Shandong Yantai Development Zone International Airport Shell Hotel. It isn't all that good, but should be at least OK, and is much cheaper (~210 RMB or ~30 USD). However, you might need to hail a taxi to get to the hotel, as public transportation is extremely inconvenient and therefore not an option. This too is bookable on TripAdvisor and Ctrip.
There are a few inns around which are cheap, but they are not licensed to accept lodgers without Chinese national IDs; personally, I wouldn't recommend stay at those inns anyway because of their usually poor hygiene.