I will try to go at both places if I found enough time
Galata Tower is more popular so what I have shortlisted so far as must are
Blue Mosque, Eyup Mosque, Hagia Sophia, Topkapi, Cistern, Bosphorous Bridge, Two bazaars (Spice and Grand and may be Egyptian Bazaar), some resting places of Sahabas, walk around the districts of Fatih and plenty of food as advised by Selim I
Possibly Galata Tower and the one you mentioned (Saphire of Istanbul) and plenty of advice to follow from this thread from every post... as I will have access to Internet to visit this thread anytime