I am touching down the airport past midnight. Is there any public transport to Dongzhimen that hour?

Hi, don't worry. Even you will arrive at midnight, the airport shuttle bus line 2 is still in service then. The night operation time for the shuttle bus is from 24:00 to the arrival of the last flight. Therefore, you can take the shuttle bus to Dongzhimen directly.
By the way, the taxi fare from the airport to Dongzhimen is about 80RMB at night.