Istanbul Ataturk airport to Sultanahmet or Taksim more less 20 km. You can book your shuttle bus from your hotel or travel agencies around Sultanahmet. (Most of the hotels offer shuttle bus service). Please becarefull with taxi drivers, most of them take you around the city and trying to charge more than what you should pay. Taxi fee from Airport to Sultanahmet or Sultanahmet to airport max. 35 to 40 Lira. Before you take a taxi you can ask price for 35 Lira, it is better for you not to get a surprise price.

In asian side of Istanbul there is another airport which is Sabiha Gokcen, If you take a easy jet, corendon, German wings or similar charter flights, you should flight to this airport. Sabiha Gokcen airport is pretty far away to Sultanahmet or Taksim area. You should take a shuttle bus to Kadikoy first than take a ferry to Eminonu. After that use tramway to Sultanahmet. This is a cheapest way to get Sultanahmet. Or you can take a taxi but it cost min. 50 to 60 euro.