What's the train/taxi fare to the airport from a hotel at Nanwanzi Bystreet, Nanheyan St Dongchen?

In my opinion, you can take the airport shuttle bus line 3 to reach Yabao Road Station and then take a taxi to reach that area. The price of the shuttle bus is around 16 and the taxi fee is around the taxi fee is around 15RMB.
If you want to live near the airport shuttle bus route, Beijing City Central Youth Hostel Hotel is suggested. It is near the local Train Station, which is also the terminal of airport shuttle bus line 3.
I think there is a good hotel in the area you mention, which is called Days Inn Forbidden City which is good, too and the transportation there is convenient, too.
The local subway is not suggested to reach your hotel.

Thanks for your information, Kate.
Could you please confirm the cost of taxi from Days Inn Forbidden City direct to the int'l airport and if by taxi / shuttle bus, how frequent are the bus trips to the airport?

Is the shuttle bus 16RMB per person and the taxi fare 15RMB is for 2 persons?

If you want to take taxi directly, the fare is around 80 to 100 RMB.
From your hotel to the airport, you cannot take bus at Yabao Road, you may need to go to Peking Train Station to take the buses and the interval is around 30 minutes.
Yes, it is 16 RMB for one person and the taxi fare is around 15RMB for two persons from Yabao Road to your hotel.