Why should approaching her be different than any other girl? Just treat her with respect, act like a gentleman, and use good manners.
She might not want to go to a movie, a bar, club, etc, but if she likes you would probably be open to going out for coffee, lunch, etc. Somewhere there are other people around and she won't feel ill at ease till you get to know one another better. Let her meet you someplace so she can leave if she wants and not feel trapped.
Christian girls do go out on dates, that's usually how they end up getting married. It's not arranged from birth.