What are the hours of operations of the 72 hours free transit office/counter at the airport?

You can apply for the free transit anytime at the customs counter in the airport.
I guess your destination is Beijing. There are many famous Wall sections near the city, such as Badaling, Mutianyu, and Juyongguan. You can find many farmhouses providing dinner and accommodation at the foot the Wall. If you only want to visit the Wall, it is recommended you live in one of them. In addition, many hotels are in the city for your choice.
Please remember to get back to the airport in time.

Yolanda, you are suggested to choose a hotel in the city area. Living in the farmhouse may not be a very good choice for foreigners because of the language barrier.