Liqun is south second ring road I believe around Qianmen Done Dajie (all of the cabbies know it and the Chinese address is easily found online). It will take (without traffic) about 45 mins each way I would say. Give it an hour.

You can be out of the airport and in a cab in 20 minutes (I've done it in 10 without VIP - never done VIP, and with the little terminal train coming right on time - but this is an exception to the rule).

Assumptions:

15 minutes to deplane
40 mins immigration, customs, getting into a cab (assuming bags are checked through - can and should lowered by 10-20 mins)
One hour to Liqun (it shouldnt be at 1PM-2PM)
One hour for eating (probably a bit shorter especially if you are alone)
One hour stroll around the hutong (probably more like 15-30 mins)
One hour back to airport (up to 1.5 - if there's traffic you can tell the cabbie to get to 4th ring road and go from there and you'll be fine).

I think I'm being pretty liberal with regards to timing estimates here and there is an hour to spare.

Lido will be easier on time, obviously (its essentially off the airport expressway) but nowhere near as cultural or interesting, IMO.

Another idea is 798 art district which is also essentially off the airport expressway, I believe (or in that area). Transport to and from will be easy, there are plenty of cafes to eat/drink and of course art. Lots of junk IMO, but also some nice pieces. I like the atmosphere in the area more than anything.

Moondog et al: If any of my assumptions are way off, please do make note of it and let me know...