Estrogen's effect on the brain is a complex story, one which we are only beginning to understand. We know it's important for women, but we're not sure about the details. One of the problems is that it appears to interact with stress. There are two aspects to estrogen's effects on women: normal monthly fluctuations in estrogen levels, and menopause.
It's also important to distinguish post-menopause (once you have completely stopped menstruating) from perimenopause (the years of menstrual irregularity leading up to this).
In general, the last few years of research seem to be coming to the conclusion that any cognitive problems women experience as they approach menopause is limited, both in time and in task, and depends in part on other factors. For example, those who experience many hot flashes may have poorer verbal memory, but the main cause for this may be the poorer sleep quality; those who are distressed or experience mood changes may find their memory and concentration affected for that reason. These findings suggest the best approach to dealing with cognitive problems in perimenopause is to tackle the physical and/or emotional causes.
Post-menopause is different. Post-menopause is all about low estrogen levels, and the importance of estrogen for brain function. Nevertheless, estrogen therapy for postmenopausal women has had inconsistent results; there has even been some research suggesting it may increase the risk of later dementia. There is also some suggestion that it may not help those women who have cognitively stimulating environments, or are highly educated. And other indications that timing might be critical -- the age at which you begin hormone therapy. At the moment, we simply have too little clear evidence to warrant recommending hormone therapy for cognitive reasons (particularly in light of the possible cancer risk), or to know when it might be effective.
Excitingly, however (because there is no downside!), there is some evidence that physical exercise can counter the cognitive decline postmenopausal women may experience. There's also a study suggesting that the effect of low estrogen after menopause is not to impair cognition but simply to change it -- however, because women aren't prepared for, or understand, these changes, they perceive it as impairment. That would suggest that what is needed is an education program in how the brain changes (but first we have to understand exactly how it does change!).