First of all: Don't hesitate and just book Hatices Saffire Apartment! You won't regret it, but love the view and all the rest ;)
The apartment is comfortable, very clean, big enough and helps you to stay in a good shape as you have to climb up 90 stairs. And of course the view is stunning. No matter how often you look out of the window.
Not to forget it's a perfect basis to explore the city and still have a quiet home. 5 min to Istikal Caddesi, 10 to Galata Tower, 15 to Galata Bridge, 30 min to the Topkapi Palace and Hagia Sophia/Blue Mosque!
Our personal advice:
Food: Have some excellent fish at the fish market just on the right side of Galata Bridge (esp. first restaurant on the right side, reservation needed).
Best Chicken Dürüm on the way from Galata Tower to Galata Bridge, right after the steep part of the road. It's a kiosk at a small place next to the bigger street, you won't miss it when walking past.
Traditional turkish Meze with very friendly staff and delicious chocolate cake (Asmalı Mescit Cd 10-26, Asmalı Mescit Mh., 34200 Beyoğlu/İstanbul, Türkei
http://goo.gl/maps/l1r7p).
And of course have breakfast at Hatices Cafe La Perla around the corner of the apartment. Omelettes highly recommended!
Must see: Topkapi Palace and Harem (pass the long queue and buy your ticket at the ticket machine, it really works!), sunset from the Asian part of Istanbul (take the ferry to Üsküdar and walk to the Kz Kulesi).
Well then, enjoy Istanbul in a great apartment. We will definitely come back!!