what is the most efficient and cheap way to see Bird Nest from Beijing airport.?

Take airport express, and get off at Sanyuanqiao. Then walk to Sanyuanqiao subway station directly. Transfer to subway Line 10 (Outer ring) to Beitucheng Station. Then transfer to subway Line 8, and get off at Olympic Park, Exit D. After arriving, walk to Bird Nest Directly. This route takes around an hour.

You just have 8 hours to tour freely, so you are recommended to take a taxi to get back to the airport.