Right outside the airport, you will see many taxi-cabs lined up. You can probably find a cab at any time of the day, and the fare will be cheaper than most of the other countries. You can use http://taksiyle.com/en to calculate how much it will cost, before you take it.
To save on money you could get a cab to Florya train station from the airport (about a 10 min cab drive) and get a train to Cankurturan (about a 40 minute train journey) which is right in Sultanahmet area with most of the hotels and hostel.
