Crossing from Galata to the Historic Peninsula is a five-minute walk over the bridge. Four major buildings in a day is possible, but the order matters; done in the wrong one, half the day is spent in queues.
Start early. Hagia Sophia and Topkapı open at nine; being inside at half past is not the same day as arriving at eleven. Walk across the bridge — and from Eminönü it is another twenty minutes on foot to Sultanahmet if you would rather not take the tram.
The order: Hagia Sophia → Basilica Cistern → lunch → Topkapı Palace → the Blue Mosque. Hagia Sophia is emptiest first thing. The cistern is next door and takes forty minutes. Topkapı wants the most time — the Harem is a separate ticket and another hour, and it is the best part of the palace.
The Blue Mosque is a working mosque; visiting hours move with prayer times and the afternoon is easier. Shoulders and knees covered; headscarves are handed out at the door.
What to drop. If time runs short, drop the cistern. It is striking, but of the four it tells you the least. Do not drop Topkapı.
The museum pass pays for itself if you will see more than four museums in three days, and it skips the ticket queue. For Hagia Sophia and Topkapı alone, single tickets are the better arithmetic.
Going back. Walking up to the Grand Bazaar in the late afternoon and down to Eminönü from there is the way to end the day at the Spice Bazaar. Cross the bridge on foot again — at sunset the Golden Horn is a different place from the one you saw in the morning.