Thank you all for the pointers.

Here is a recap:

We arrived just before 5am, got the 24hr arrival visa by 6am.
Nothing open in the ghost town that is Terminal 2. No Internet connection either.
A taxi driver offered to take us to Great Wall Mutianyu for 500rmb, but I had no idea if that was a fair price or not. Tried to get money from an ATM and failed.
We ended up taking the free shuttle to T3, with the idea of finding more info or taking the train into town (didn't realize it goes to T2 too).

At T3, a tour counter was open, and we paid $140 (credit card) for a roundtrip to Mutianyu. It took about 90 minutes each way. We had a nice 2-hour hike (up and down and up and down). A few drops fell but stopped, so the temperatures weren't too hot. Views weren't great but nice atmosphere. Not that many tourists (10am-12pm) but more were arrriving as we were leaving.

Saw about 200 police/soldiers lining the highway over 10km. Apparently a few of the African dignitaries visiting Beijing were also at the wall in the morning.

We got back to T2 by 1:30pm. Plenty of time for our 6pm flight, though the security system at T2 was incredibly slow (hand checks for everyone). We had too much time it turned out, as the only Priority Pass option was closed. Nothing but a Starbucks and tiny place with overpriced, mediocre craft beer and frozen pizza. We should have gone to T3 for a decent meal.

All in all a good transit. Only disappointment was we missed out on eating any Chinese food while in China.