Scanned from albumen photo attributed to Athanasiou. Estimated date is around 1880: the Frankish Tower has gone, but Elgin's wretched clocktower is still standing (to the right of the frame). This makes the range 1875-1884.
Public domain image.
This building was demolished in 1914. It had originally been built as a Medrese (theological school) in 1721. By 1821 the Ottomans were using it as a prison and hanging Greeks from the old plane tree (destroyed by lightning in 1919). The Kingdom of Greece continued the barbaric Ottoman practices and the prison at the time of this photograph was notorious for the inhumane conditions of incarceration and for the hanging of prisoners on the plane tree, which were seen by passers-by.
The prison was closed in 1911, and all but the ornate Ottoman entrance of the Mendresse demolished in 1914. The entrance is opposite the Tower of Winds, on Markoy Aureliou Street.
From the architectural design of the building, I am now confident that this is actually the former Medrese.