I have been informed this entrance is located at 36.197500 N, 37.161228 E; it appears to be the entrance to the Khan ash-Shuna (Shouna), a late 16th century (according to one of my travel guides) Ottoman caravansarai opposite/north of the Khosrowiya (Khusruwiya) mosque. Between the two runs the Suq ash-Shuna, as you correctly surmised. The khan seems to have been restored in 1990.