As the title says, the replay function does exists in Naruto Online, but it's a bit hidden and it's not even translated but it works. Those who watched Kaizer's video already know about it, but his videos, unfortunately for us, got removed so here is a alternate video I made, a short guide how it works.
https://www.youtube.com/watch?v=ed140*jE
The base is that you can see or save your last fought battle, when you start a new one your old replay gets replaced unless you save it on your computer.
Everything is explained in the video. It's easy, just no one told us about it ( just like with the recent new function for ninja tools :) )