proTeXt uses MiKTeX, so basically you may follow this answer which shows how to install additional packages "by hand" with MiKTeX. However, because you don't want to install a package file (.sty), but a BibTeX style file (.bst), you should replace the recommended subdirectory tex\latex\misc with bibtex\bst\misc.
A more important caveat (and the the source of your problem): The current version of the natbib package doesn't include a natbib.bst style file anymore. Googling for natbib.bst brings up this file from 1994 (!). It could be that natbib.sty (the package) is backward-compatible with natbib.bst; it could also be that the package and your .bst, even when installed correctly, don't play together well. So: check thoroughly if everything in your bibliography and in-text-citations is working as expected.
Using TeXmaker's (any other IDE's shortcut keys) will not speed up the basic compilation process. However here are a few items for you.
1) If you are in the midst of just editing text then I suggest that you just use F6 then F7 to compile and view pdf OR use F2 then F3 for compile and view of dvi. Caution: The exact placement of items on the page may change when the proper sequence is run.
2) If you have added a new reference, then you must run F6, F11, F6, F6, F7 in that order to resolve all the references. (Note: If you are using internal citations to your own text, creating an index or glossary, then there may be additional steps for the complete document preparation.)
3) There are another options if your paths and installation are properly setup. You could use either texify or a make utility appropriate for your Operating System. (Note: If you are new to LaTeX then I suggest that you stay with options 1) and 2) as needed.)
When I was teaching LaTeX to college students, I found this to be confusing for the students. The most useful guidance I found for new users was 1) to compile and view as a normal document. And then periodically run 2), the full sequence. (I advised that the first and last compile for a session when working on a document would be F6, F11, F6, F6, F7. (pdflatex, bibtex, pdflatex, pdflatex, pdf viewer).
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