There is no direct bus could be used. You could take a taxi from the South Railway Station to get to the airport in one hour with a fare of about CNY100.

Alternatively, you could take the bus 135 from south railway station to reach Changsa Railway Station firstly, and then please transfer to bus 114 which could drive to you Changsha Airport directly.

At the south railway station, you could also take the bus 16 to get to Changsha South Bus Station, where you could take the airport shuttle to Changsha Airport.

Of course, the latter two methods will take you a little longer time on the way, about 1.5~2 hours.