Does high school success in Math have a correlation with Undergrad success in Mathematics?
You are who you are. Don't be overly concerned about others. You will need to work very hard, of course, but the fact that others may be able to pick things up more quickly than you doesn't mean that you can't excel. 

The fact that you were admitted suggests that others have faith in you. Have faith in yourself. 

You can also try to find a study group with others at the new place so that you get additional exposure. Use good study techniques and never - never fall behind. Start early on assignments. Ask a lot of questions. I was once thought to be very smart because I asked a lot of questions in class. But in reality I just wanted answers that weren't apparent to me. 

You might also try as much as possible to limit outside interests until you find your way in a new environment. However, agreeing with a comment by user aeismail, that doesn't mean to give up everything for academics. Spend several hours each week on something that contributes to your health, reduces stress, and engages your mind in a different way. I now use Taichi which does all of these things. As an undergraduate I practiced Judo several hours a week. I didn't get very good at it, but it was a great complement to studying. As a grad student I rode a bicycle with a local club - fast but not competitive. That was physically satisfying, but didn't have a mental element like Taichi does. But don't occupy yourself with trivia.

But yes, studying great books in advance ain't gonna hurt you any. I studied out of Rudin over 50 years ago and still have the book(s). 

You might also find that some courses are more challenging than others Analysis v Algebra for example. That is natural as the core thinking processes and insights are just different enough that you can groove on some things and find others a bit more work. Put your best efforts where they do the most good. 

Talk to your professors. Ask questions. Listen. Ask more questions. Correlation only matters for groups. You are who you are. 



Why you ask questions:

Here is a story (real occurrence) from the University of Kansas about 50 or so years ago. A prof, quite well known and respected, was lecturing in Topology. This meant writing proofs on the board. Occasionally a step wouldn't be filled in with the statement "Of course it follows, trivially, that ...". On one such statement, a student didn't see the obvious connection and so asked. The prof looked at the board and the developing proof for a few minutes. Then walked over to the corner of the (chalk) board and started making notes to himself in a tiny script. He went into totally abstract thinking mode, ignoring the class. After a bit of writing and erasing, etc, he wandered out of the room toward his office. The students followed along, gathering outside the office. The prof started pulling books off of his shelf and consulting them - several books - several more minutes. 

Then he seemed to find enlightenment and returned to the classroom. When everyone was again seated he announced. "Yes, of course. It's trivial."