Lucky!(?) I got the Jinhangxian International Business Hotel, which I would not recommend, especially if you're planning to do any exploring in the city. As others mentioned, if you're flying economy they pair you with someone in the room.

My plane pulled into a remote stand at around 18:00, and I cleared immigration and customs by around 19:00. Walked up to the Hainan Transit counter in T2, and the agents there had my name on a handwritten sheet of people presumably eligibile for the free hotel. There were a total of 8 people. The hotel group did not depart (walking to the shuttle in the parking lot) until around 19:30, and we arrived at 20:09. The hotel is in the middle of nowhere, with hardly anything nearby, and I imagine it would be difficult/expensive to get into the city from there. Initially we were only given one key between my roommate and I, but I was able to ask for another. The room itself was ok, but smelled funny (a little sewagey), had no A/C, and the carpets were stained. The bed linens were clean, though. The shuttle back to T3/T2/T1 was a small overcrowded minibus and people were climbing over each other inside; the ride took about 20 minutes to T3 and 30 minutes to T2.

In the future I'll get a hotel in the city on my own dime.

The transit hotel I got:
Beijing Jinhangxian International Business Hotel Parking Lot
8 4th Latitude E Rd, Shunyi Qu, Beijing Shi, China

https://maps.google.com/?cid=45731302449450265

I was able to also "use" the transit lounge which is landside near the check-in counters in T1. They had satchels of coffee and tea some peanuts, but that was it for drinks/eats. There were two massage chairs, but you have to pay for them with an app. I was hoping to leave my carry-on bag there while I went into the city, but despite there being a baggage room, when I asked about it the attendant flatly said, "no, all must keep your bag". I ended up paying 40 yuan to the left luggage desk in T1.