טקסט מקורי - יפנית - 大活躍見せてやる

Does this mean:
"I'll be really successful"?

CC: IanMegill2

Well, this is only an approximate translation. The literal one would be:
I'll show you (them? someone? the Japanese has no object) what a great role (a function, involved in some great project, for example) I will play.

So, would you say that this translation is acceptable?

"I'll show how successful I'll be" ?

Yup, that's a closer approximation, anyway!