Well probably introduction to analysis is intro to real analysis. Real analysis is just analysis done only on the real line (as opposed to complex analysis or vector/multivariate analysis which works in the complex plane and R^n (not sure how to write that in a forum) respectively). Basically, what you do in this course is build up a lot of machinery with rigorous definitions of limits, and then use that to prove most of what you learn in a basic calculus class. It HAS to be offered anywhere there is an undergrad math major, so it's probably just not called real analysis. You'll certainly know once you see your textbook, but my guess is that the 332 course is that one (and very likely NOT the 439 course). Probably the 632 is the masters level version. Definitely do NOT take that until taking the intro level course. Math is one subject where I don't advocate skipping anything to move onto something more difficult.

As far as how hard the course work is, well...I'm not sure exactly how to answer that. I've found first year courses fairly easy, though of course you do have to put in a reasonable amount of work. People with less of a math background generally seem to find them harder. It's certainly not something that is unbelievably difficult.

To get into the schools you are talking about, you should be reasonably fine. Those are good schools, but a lot of what you see on places like econphd is geared for people who are shooting for a top 5 school. If you do well in your analysis class, reasonably well on the GRE, maybe work in a few more math classes and get good letters, you should be in very good shape for the kinds of places you are looking at. In fact, if you do these things you will want to consider at least applying to some of the top 10/top 20 schools (unless you would not want to go to them).