How do we book a taxi from Xiangyang (Xiangfan) airport to Wudang? And how much does it cost?

It is hard to book in advance, unless you ask for a pick-up service. It is suggested to hire one when you reach the airport.
The taxi fare is around 400 to 450RMB for hiring one.

Thank you for the reply Lily. Do you know how to arrange for a pick-up service, as we may be pressed for time and it'd be easier if we already had something in place?

Actually, it is a little hard to ask for the pick up service from the airport to Wudang directly. Therefore, I suggest you take a taxi to XY Railway Station, with the taxi fare of CNY40-50. Then you can easily catch a direct bus to Wudang outside of the railway station.