Either weeday or weekend, you'll have enough time to make it to the city. As far as oddities, the Beijing Underground City is pretty fun. It is a network of tunnels built during Mao's paranoia about war with the USSR, and it quite extensive.

During a weekday, the traffic may be pretty tough - but still should take about an hour (and cost 100-110 RMB, or $13-$14) The name in Chinese characters should be ”地下 成“ Its pronounced DiXia Cheng (or phonetically, Dee-Shee-ah Cheng) . the address is is 62 West Damochang Street, Qianmen, Tel. 6702-2657, 6701-1389. I don't know the charachters for "Damochang" - perhaps someone else could chime with better chinese skills than I.

You're also in a great location to walk north through tiananmen square, then into the forbidden city for an hour or too. I'd head back to the airport by 1:30 for a 4:00 flight if you already have a boarding pass. That time of day isn't incredibly busy for Beijing international departures, and it should be about 20 minutes from the time you get out of the cab until you get through security, assuming you already have your boarding passes.