Topkapi was the seat of the Ottoman court for nearly four hundred years. It is not a single building but a sequence of courtyards, each one more restricted than the last, so that walking through it means moving from public ground towards the private world of the sultan.
The Harem is ticketed separately and worth the extra step: tiled corridors, the apartments of the valide sultan, and rooms where succession was decided. In the Treasury are the objects the empire kept closest, and from the terraces at the far end the Bosphorus opens up below the palace walls.
Half a day is realistic. Go early, and buy the Harem ticket at the same time as the main one to avoid a second queue. From Hotel Sultania the main gate is a short walk up through Gulhane or along the tram line.