Tonight, the moment I hit the outside stairs, the skies opened up and the deluge began. I couldn't believe my mazel with the timing. :)
It wound up being more fortuitous than I had expected. Within a few minutes of my entering the apartment, I heard the sound of water rushing. The "little" room (aka where this laptop and my brand new printer sit) was flooding, courtesy of water pouring through the window frame. I spent a frantic 30 minutes moving everything around (most things got soaked, but B'H' I managed to spare the electronics and their cables) and mopping/sponging up the water.
I also placed a few calls to my landlord. He dutifully came over, and I heard him climbing up the ladder en route to the roof. By that point, the water had stopped coming in so fast, and things were in hopes of beginning to dry out shortly. I met him on the outside stairs when I was returning from throwing out the schmattas I used for the cleanup. He told me that people had been on the roof, and that a cup the exact size of the drainpipe had gotten caught in the pipe, causing backup. I joked that I suppose Hashem wanted me to be holier, because the little room now qualified as a mikvah. But I felt sorry for him, because you could tell it was really eating him up that people had been up on the roof. I gather nobody save him is permitted up there, given that the roof isn't fenced off, etc.
Things finally dried up a little while ago, and I shoved everything back to their respective places (mental note: in future try to avoid obtaining more heavy stuff). All in all, quite an eventful evening, and hopefully one that will not be repeated, for everyone's sake.