My flight arrives Changsha Airport at 21:00. How can I take a shuttle bus to west bus station?

According to your landing time, there is no direct shuttle bus that can take you to the West Bus Station. Thus it's suggested to take route No.1 and get to Civil Aviation Hotel (close to the Railway Station). It takes around 40 mins on the way. The ticket fare is CNY 16.5 per person.
After that, take a taxi to the destination directly. The cab fare is CNY 30 to 40 at night.