That's not a black eye, it's a representation of someone going pale with shock or surprise. Granted, it's normally across both eyes, or at least a bit more extended across the face, so the confusion in this instance is understandable.
This is indeed humourous and Yukari is being a tad cruel to her. According to the artist comments, Ran is supposed to be babysitting Yukari and, from what we can see, Yukari doesn't like that.