1. Take a Bosphorus Tour
A cruise along the Bosphorus can give you the feel of the ancient wealth of Istanbul in the shortest time--you can do it in as short a time as an hour and a half.
2. Süleymaniye Mosque
The Link goes to a very well done virtual tour of the Süleymaniye Mosque. You'll want to see it yourself, of course. Pixels cannot do it justice.
3. Egyptian Spice Bazaar
Lots of travelers complain that the Egyptian Bazaar is becoming a tourist trap, but it's still impressive to walk the aisles despite the influx of jewelery and other tourist junk.
4. A walk around the alleys of Beyoglu
Go north across the Golden Horn and you'll come to the art and night life center of Istanbul.
5. Meze and raki in Nevizade Pasaji (Beyoglu)
More night life, grab an outdoor table and order something small.
6. An evening in the fun bars and bookstores in Kadıköy (on the Asian side)
Take a trip to Istanbul's Asian side; the Byzantine Chalcedon, Kadıköy is a comfortable middle-class neighborhood with bookstores, taverns and tea gardens as well as the opportunity to take a ferry cruise.
7. Kariye Camii/Chora Church
Kariye Camii or Chora Churchis considered to be one of the most beautiful examples of Byzantine architecture in the world. It's covered with mosaics and frescos.