Can I make reservation for taxi from Sunlitun to Beijing Capital Airport around 5 a.m?

Yes, you can and the you can call the two numbers: 96103 and 96106. I remember that they have English service. If you cannot find the English service after you make the phone call, you can ask the hotel staff to help you to book one.
The taxi fare is paid to the taxi driver. Once you make a reservation through the number I give, about 3 RMB service fee needs to be charged.
You can book 12 hours before.