Here's another budget sightseeing option. Bucoleon Palace is the ruined remains of a Byzantine palace, and there isn't much left here. The walls alone are all that remain of this once great residence.
Finding It
Wander south from the Sultanahmet in the direction of the Marmaris. You will eventually have to pass under an underpass. Bucoleon Palace faces the highway and the Marmaris.
On the way, you will pass through some very quiet neighborhoods that don't see tourists everyday. It seemed safe to walk here.
Personal Experience
We tried to find Bucoleon Palace. This involved wandering outside of the tourist area, through a very poor area - well, to my eyes - I imagine it may have been a middle or working class neighborhood. The houses were poorly constructed, with bits of odd metal and no real windows, part original house, part scrap - but...it seemed like a safe area. We wandered around - found a mosque that used to be St. Sergius & Bacchus. Its graveyard was overgrown and we spent some time in that and its courtyard, peered in to the mosque from some windows. Next to it was a huge abandoned building, and after walking uphill, we realized it was an abandoned mosque, complete with a broken minaret.
Following down to the water (Marmaris Sea) at last, we found Bucoleon Palace. Facing the highway, it is built up against the sea wall and through its extant marble window frames, one can see more of what I would call "slums". Imagine, looking out each day onto an ancient palace! Not much remained - cubbyholes that may have been towers, where homeless people had moved in. Remnants of their meals and alcohol binges. Two men drinking near the other side of the wall.