As you know cappadocia is not a city name, Cappadocia is an ancient area in Central Anatolia which is incorporates the provinces of Aksaray, Nevsehir , Nigde , Kayseri and Kirsehir cities. If you want to Cappadocia area of Turkey you can use Kayseri Erkilet Airport or Nevsehir Kapadokya Airport. Nevsehir kapadokya Airport is 25 km far from city center and Kayseri Erkilet Airport is 5 km far from city center.
There are so many Airport Transfer (Shuttle) services at the airports (Taxi, bus).
Cappadocia Airport Codes
ASR Erkilet International Airport – Kayseri
NAV Kapadokya International Airport – Nevsehir
You can find flight informations official web sites of that airports.
If you will go to cappadocia from Istanbul don’t forget that there are two international airports in Istanbul, one of them is Ataturk Airport and the other one is Sabiha Gokcen Airport. Sabiha Gokcen airport is very far from central Istanbul but there is a shuttle service from Taksim square, Ataturk airport is not so far from central Istanbul.
If you have any question about cappadocia airports, you can ask here.