Not well-covered, but covered well. Or maybe this old, old well is well-covered. In fact, I asked around if people really knew what this is but most did not, and guessed a well. I see now that there is a plaque on the stone wall, so next time I walk by, I'll check it out.
UPDATE I found out that this is a sebil. Wikipedia says: "A sebil or sabil is a small kiosk in the Islamic architectural tradition where water is freely dispensed to members of the public by an attendant behind a grilled window."