Metal in-roof gutter condensation

I'm in a rental apartment have a condensation problem. It's one of the "penthouse" apartments, so the roof is directly overhead. We have a leak in the ceiling. I opened the ceiling, and here's what I found: - the roof is a flat built-up roof, circa 1890. Fairly heavy wood decking with about an inch of accumulated roofing materials. - there is a metal gutter cut into the roof deck that drains to a common drain that runs down the party wall between our building and the adjacent one. It looks like the gutter was a renovation project sometime in the past. - the gutter happens to run close to a skylight shaft. - the apartment was recently renovated, just before we moved in. I don't know conditions before we occupied. - there's a new drywall ceiling attached to metal hat-channel which is attached to the existing wood roof joists/rafters. - there's a blanket of unfaced fiberglas insulation laid onto the drywall. The sidewalls of the skylight shaft are not insulated. - there's no venting for the roof. The underside of the metal gutter is coated with water which is dripping into the insulation, which then is accumulating and leaking through the ceiling. The past couple of days the weather has been reasonably mild and the underside of the gutter dried off. Last night it snowed, and the gutter is now sopping wet. My research suggests that the most straight-forward solution is to spray the underside of the gutter with a foam insulation. Am I correct? Are there other possible solutions? I need something straightforward to tell my landlord.