what is the best way to get to xin xuang ya yuan apartment from beijing capital airport at 6am?

But the apartment has more than one branch. I presume you refer to the Wangfujing/Xinshijie branch and here is the way:
take airport express train to Dongzhimen. The first train leaves from T2 at 06:35 and T3 at 06:20. At Dongzhimen, you can change to subway line 2 to Chongwenmen. Get out of exit G2 and walk south for 5-8min to find the apartment.