The only problem with your solution is that it would very likely increase the rate of murder, probably much more than it would decrease the rate of rape. (At least in EU and other countries where weapons are generally banned.)

If you want to reduce rape right now, you should start by putting more effort into getting rapists in jail. Currently three fourths of the rapists get away scot free. Get the police to do their job, don't force everybody to be their own police.

We have many jurisdictions in the US where open or concealed carry is legal, and even someone with an unlicenseed gun would likely face only a minimal sentence on a weapons charge if it were used for self-protection in a violent confrontation. While Google does turn up some reports of women shooting would-be rapists, it doesn't seem to be that common. We could generalize fromt hat to observe that we still have other forms of crime like mugging, assault, etc. despite the widespread availability of guns, and at considerably higher rates than many other developed countries.

Lastly, you only seem to be considering the sudden-assault-by-stranger sort of rapes, whereas many rapes (and other crimes) are committed by someone who has established a degree of trust, either by default (as a relative or some other social context) or by design (as in date rape or some sort of 'long con' predatory behavior).

The problem with your approach is that it can be used to sideline people who choose not to carry weapons, as well as normalizing the carrying of weapons in every context - 'shouldn't woman take guns with them into the shower, just in case? Isn't it a bit irresponsible to leave oneself defenseless in the shower, where it may be hard to hear someone break into the house?'

One could equally suggest that every woman get herself a large dog. People are extremely respectful of personal space if you step out with a German Shepherd; my neighbor observes that she feels quite safe taking a nap at a park or beach in the company of her dogs. But just because anyone could do this doesn't mean everyone should do this.

There is but a small problem, nothing prevents any soon-to-be rapists from getting guns themselves.

Nice victim blaming - they got raped because they didn't want to shoot someone.