I usually ate in the Taksim Sq or in the old city. There is a busy street right next to the hotel and it has many restaurants, including MickyD and Burger King, Pizza Hut and KFC. About 2 blocks down the same street you'll run into some side streets with numerous restaurants, shops, etc. Mostly locals hangout in that area so local food galore.

Sultanahmet - Take a bus to the first tram station (the name escapes me at the moment) and the tram to the old city. 20-30 minutes total and 2.60TL per person each way.
Taxi is the other option but I have no idea how much it would cost.

To/from airport - Due to delayed BA flight and extremely long immigration lines I did not get out of the airport in time to catch the last shuttle(12am) to Taksim Sq., so I had no choice but to take a taxi. It was past midnight so an extra 50% is added to the regular fare. It was ~$42US. The extra 50% is added between midnight and 6am.

This morning (6am)I'm catching a taxi from the hotel to Taksim Sq. I was told it will be ~10TL. Then the shuttle to the airport at 9TL, or a total of ~$17-18US. The shuttle takes about 40 minutes and leaves every 1/2 hour.

Hope that helps.

Checking out in few minutes and I'm on my way home.


EDITED TO ADD THE FOLLOWING:

Taxi 6am - Taksim Square 6:10am - 10.50TL

Shuttle dep 6:30am - Airport arrival 6:52am - 9TL (I guess the 40 minute estimate is for the rush hour traffic times, as our driver was flying that full size coach)

Security, check in at the BA F line, (if I had to use the Y queue, it would have been at least 1/2 hour longer), passport control, total - 20 minutes.

Walked into the BA lounge at 7:20am - way too early for my 9:10am flight - oh well.