I don't know anything about Apostol's Mathematical Analysis. My guess would be that it demands a fairly sophisticated background of the reader, and does an excellent job of covering calculus from an extremely rigorous point of view.

I have heard that Apostol's Calculus is an excellent choice, probably somewhat more accessible to beginners, but still offering a rigorous, highbrow perspective. I've also heard the same of Spivak. I'd probably opt for one or both of these.