I just returned from Turkey. There are many new bus routes to various parts of Istanbul from the new airport. These are Havas Buses, which are large comfortable buses (think Greyhound). Price to points in the Old City (Sultanahmet area) are 18 Turkish Lira. You can pay with an Istanbul Kart (public transit card you load money onto), or sometimes by credit card. I used the Havas Bus three times and never had to wait more than 10 minutes for a bus. The driver will place large suitcases in the hold of the bus. Very convenient! The ride to and from the Old City is about 45 minutes to an hour, depending on traffic.