Just drive better..... you could never beat a type-r if the guy has the same skills like you. The reasons why it's that hard to beat a type-r is the good balance setup by the factory, but not on a civic non R

You could do mods on your civic but the type-r could do the same thing also, so just try to squeeze out everything on your car and you'll be fast enough no matter what car are you driving. This is my thoughts