Well, yes, menopause can cause bloodshot eyes. As we know that menopause is a natural biological process for women. People with menopause may suffer some eye problems, such as dry eyes. But on the other hand, there are some other common symptoms of menopause, such as breast cancer, stress and anxiety, high blood pressure. So when you have high blood pressures, it is likely to get bloodshot eyes. That is why you get bloodshot eyes, when you have menopause. For your situation, I think you can use cold ice to cool your eyes. And then just go and see and eye doctor and take an eye exam. Also, treating the menopause is also essential.