From Beijing international airport to Great Wall of China at midnight

When you land at 3:00 AM, you should probably take taxi. It'll cost you around $55. (I've checked the rates over a site that i own, so i'm not sure if i should share the URL. Because i don't want to get marked as SPAM)

When you land at 11:00 AM, you should probably take the bus and following are the choices taken from http://www.beijing-airport-transportation.com/beijing-airport-to-great-wall-of-china.html

Choice 1: Take Bus No.359 and get off at Zuojiazhuang for the switch of Bus No.379, then get off at Xisanqi stop for the switch of Bus No. 919 to Badaling Great Wall.

Choice 2: Take Airport Express Train to Dongzhimen for the switch of Subway Line 2, then get off at Jishuitan Station, walk to Deshengmen and take Bus No. 919 to Badaling Great Wall.

Choice 3: Take Airport Shuttle Bus Line 5 to Baofusi Stop, then take Yuntong 109 to Xisanqi Stop, finally take Bus No. 919 to Badaling Great Wall.

I haven't been to China yet but i'm sure that a sunrise at Great Wall would be spectacular.

I hope it helps.

Beijing airport shuttle starts from 5.30 AM, and there will be no buses at that time. You will need to take a taxi.

I haven't been to the part of Great Wll you mentioned but I've been to Badaling and Mutiyanyu. Both are about half an hour from a bus, but again, you will have to take a taxi. Even with a bus, you will have to take 2 buses to go from Beijing (say, Tianmen East) to the Mutiyanyu, so your better chance is a taxi.

I doubt you will be able to view the very first scenes of the sunrise, specially since you have to take a ticket and booths usually open at 6.

Note that the sunrise in Beijing is usually at 5.30 AM. Also, there is a nice park just outside the Tianmen square, "Beihai" park, that many locals go to see the sun rise. 2 Yuan entry ticket. After the sun rise, you can even go back to the Tianmen square to see the flag raising ceremony.