The question is pretty straightforward. Can a male writer write a first person story of a female protagonist? What limitations might he face in writing about the other gender? Should he just stick to male protagonists? Is there a book written about the scenario mentioned? If so please advise.
There's no inherent reason why an author can't write from a perspective that does not exactly match every characteristic of him- or herself.
Yes, trying to write from the point of view of a member of the opposite sex creates challenges. But so does writing from the POV of someone of a different nationality, or religion, or political persuasion, or occupation, or age, etc.
Presumably almost all fiction stories include characters who are not exactly like the author, so a decent writer has to have some ability to depict people different from himself.
In some ways I think it would be easier to write from the POV of the opposite sex than some of the other things that make people different. Personally, I know plenty of women. I talk to women every day. I have lots of opportunities to hear things from a female perspective. I know a lot more women than, say, black people. (I'm white.) I very rarely talk to people from other countries. I've only known one Hindu in my life that I recall. Etc.
I think I'd have a much harder time writing a believable male atheist farmer than I would a female Christian software developer.
There is no problem. Random Acts of Senseless Violence by Jack Womack, Friday by Robert Heinlein and many of his short stories, Contact by Carl Sagan, Big U by Neal Stephenson, Cybernetic Samurai by Victor Milan. It's quite common in speculative fiction and fantasy anyway.
The observant writer shall have no problem. The unobservant one will pointlessly limit the possible scope of their narrative.
Just like female authors often write a story with a first person male protagonist.
Authors who socialize and are well informed about the world lose any wrong ideas fast. For instance, out of three engineers today I spoke with, two were female. If a male author presumes that his perspective character is very unlikely to be an engineer because she is female, he'll end up unnecessarily restricting himself in terms of what can occur in the story. It may end up worse than unrealistic: boring.
As Cordelia Fine writes in Delusions of Gender, well controlled experiments show that there are no gender based differences in thinking that are not conditioned, or which are not due to mistaken beliefs and stereotype fear.
There are only slightly different preferences, not abilities, in childhood, due to slight sex differences in the brain. These turn out not to be sex differences that affect abilities.
So nothing prevents an author whatever their gender from writing in the perspective of the other gender, if they are well informed about a wide variety of life histories.