The term "B movie", at least in modern movies, doesn't have any kind of official meaning. It's not a rating that gets applied to a movie by some organization, it's really just a "genre" of movie.
Originally, the term B-movie was used in much the same way as the B-side of a musical single record; the "B" movie was the "other half" of a double feature. The B movie was use as the lead in -- the same place we currently see trailers, for example -- and rarely got much promotion of its own. These days, double features are pretty rare, so the term almost never has this meaning.
Instead, it's typically used as a reflection on the budget the movie has. A B movie is a low-budget movie that is not an "art" film. The term isn't always meant as an insult -- sometimes B movies become very well known for taking risks or breaking movie conventions that high budget films would never dare. Just because a B movie becomes popular doesn't change the fact that it was filmed on a low budget, with all of the associated stylistic choices that come with it.
So no, there's not really any way for a B movie to "become" an A movie, any more than an action movie can "become" a horror; it's just a way to describe the way the movie was made.