Arriving at the international airport, you first have to take a free bus to the nearby airport for local flights. Otherwise you might walk the 5 minutes if you don't have too much luggage. At Sultanahmet you can find cheap places with dormitories if that's what you're looking for, but they can be crowded. A middle class hotel recommended to me at Sultanahmet was the Hotel Aya Sofia , Yerebaytan Caddesi, 33. It's a small hotel with 10 rooms, clean, quiet at 100 metres of Aya Sofia.
Rooms have their own mandi with hot water.
Breakfast can be had next-door. A but higher on the scale is the Hotel Ararat , Torun Sokak, 3. It's right next to the Blue mosque. There's a new pension which was recommended, the Cordial House Pension at 40 metres south of the Cemberlitas tram stop. They also have internet access.