I know! Talk about adding insult to injury. At the very same time that we start losing our thick tresses, we start growing hair on our faces. This is another side-effect of normal, hormonal changes during menopause. The male-type hormones (androgens) you have fluctuate right along with the female-type hormones (estrogen and progesterone). Often, facial hair becomes thicker and coarser as well. You should mention this to your doctor just to monitor the changes and rule out other causes.
Options for getting rid of it include plucking, electrolysis, bleaching, shaving, and laser hair removal. There are drawbacks to each—be sure to do your research before trying them.