Do not try to go to Xiangshan (or Botanical Garden which is in the same area) on an October Saturday. Traffic jams are a nightmare, with half of Beijing trying to get there to see the autumn leaves.

Panjiayuan only if into markets/shopping. 798/Caochangdi Art Districts if you're into that.

--Central area: Zhihua Temple then Ancient Observatory then Dongyue Miao. Onsite needs are 1 hour, 1 hour, and 1.5 hours. Transport between the first two could be done on foot, then subway one stop up + a walk between the 2nd two.

--Near SW side: Fayuan Si, Ox Street Mosque area, then Tianning Si (oldest building still left in Beijing, Liao dynasty), Yuyuantan Park, and either Military or Capital Museums. All of this could be done in a circuit fitting your time frame.

--The newish Automobile Museum, SW part of city. If into cars.