I normally pass on hotels comped by Chinese airlines because they are often awful places and tend to be near airports (i.e. not much else around), but if CA offers you the Hilton/Langham or --more likely-- Crowne Plaza, I'd be inclined to accept, especially if you are allowed to check in at that early hour. Basically, you'll be able to relax during the morning, head into town clean and refreshed, and then you could drop by again several hours before your next flight. If the hotel on offer is not one of the 3 I mentioned, I'd want to think more, but would likely pass.

Regarding your day in Beijing, the plan I suggested in the "one day in Beijing" thread might be old, but the basics remain unchanged: Made in China --> Forbidden City --> Beihai --> Houhai/Gulou (there is a Great Leap Brewing place near Nanluoguxiang)