And of course he's blazing fast.
It isn't that hard to make car faster than his, but the problem is the SP bar. At the speeds you'll be driving at, the penalty you'll take for scraping the wall is enormous. So finding just the right setting balance to be able to keep up with him AND stay off the walls is pretty challenging.

Tricks like waiting 'til your tires warm up or following him until you find a spot on the C1 favorable to you all come into play, like they often did in TXR. But that's just the SP battle. You still gotta beat him in DP and RP Battles to...

I just got the game and realized how difficult it is to do DP and SP battles when you reach A class..

What car are you driving? On some tracks it's easier to do DP battles than others (Suzuka West is a b****).
Try staying as close to the inside of a corner as possible when doing DP battles, that way you'll get a lot more points. Also, try to keep your multiplier up, it's a life-saver!