Is hair loss common during menopause, and are there any ways to prevent it?

Female baldness. Remember that baldness is genetic and affects both men and women. It could be that you have inherited a gene for baldness and are just now seeing it. Rogaine (minoxidil) is the treatment of choice for both men and women if this is indeed female pattern baldness.
Yes, take estrogen. The hair loss that is due to menopause is due to decrease in the estrogen levels in the body. In women this decreased estrogen results in an increase in testosterone relative to the estrogen. This elevation of testosterone causes the hair loss. Other things unrelated to menopause but happen during middle age years of life can cause hair loss such as vitamin deficiency and thyroid deficiency.