I called the complex's off-hours emergency line, explained the problem, and they said they'd send someone to fix it. While I waited, I cleared out everything from under the sink, because I knew that the plumber would probably want access. It's a good thing I did that for a different reason, as it happens.
After a while, the plumbing company called. They had a man in the complex already, working on another job. (This complex is large and has not-great plumbing. Above All Plumbing seems to have someone out here every day, as a nearly-full-time contractor.) They told me that when he finished that job, he'd come over and deal with my drains.
I killed time on my computer. Eventually, I realized that I heard water running. I'd at first thought it was coming from outside, but going into the kitchen, I realized that what I'd heard was my kitchen sink overflowing and dripping out all over the kitchen floor. One of my upstairs neighbors had run his dishwasher and it was overwhelming the clogged drains, to my expense.
I ran for a bucket and started bailing the kitchen sink into the toilet (the bathroom drains continued to run just fine). Looking outside, I saw that the plumber had arrived. I ran out and told him, "Hurry! My kitchen is flooding!" He ran upstairs and told the neighbor to please turn off his dishwasher, then came back down and got a wet-vac to suck up all of the water.
Fortunately, I'd cleaned out the sink (see above), and there wasn't anything under the sink or on the kitchen floor to get water-damaged, and the amount of water that overflowed wasn't enough to make it to the dining area carpet. Once the plumber got the water vacuumed away and mopped up, he worked on clearing the clog ("a grease clog," he said), everything was fine again. Once the drains were flowing cleanly, I went upstairs and told the neighbor he could use his kitchen plumbing again. He apologized for the flooding. I told him not to worry; it was my fault for forgetting to tell him that there was a pending clog and that his kitchen flows into mine before draining.
In a way, it's a good thing that happened last night and not over the weekend. In a few minutes, I'm heading off to Fernley for the weekend, and had the clog hit then, the amount of water damage could have been substantial.
Since I expect to move in April or May (earlier is better, but depends on me getting an opening in the apartment building I've spotted into which I want to move), this should be the last time I have to deal with this periodic plumbing problem.