The obelisk is from Karnak in Egypt and commemorates a victory in a 144o BC battle. On the base emperor Theodosius I (379-395) is depicted, with his court, as well as scenes of erecting the obelisk, which was done in 390 AD.
The current Türk ve Islam Eserleri Müzesi is in the former Palace of Ibrahim Paşa, a vezir of Süleyman the Magnificent who was executed. You see it behind the obelisk.