As for the Eternal Champion of Arena, he/she simply vanishes without a trace. Even in books, they are simply mentioned as the Eternal Champion, and no supplementary detail is provided to give them any form of epilogue.
It's very funny, because the hero of Daggerfall was sent on a very secretive mission (and died in some outcomes of the Dragon Break), the hero of Oblivion was pretty much eclipsed by Martin and the fact Oblivion had a boring storyline to begin with. The Eternal Champion, however, was praised by Uriel Septim himself and was to,
"take thy rightful place at [the Emperor's] side". Surely his/her journey, spanning a decade and saving the whole of Tamriel from Jagar Tharn's rule would be one of the greatest achievements of all the heroes, at least from the Empire's perspective. I always saw him/her as the most revered and celebrated, the one who should have statues all over the place. So it is kind of odd how (s)he is rarely referenced, but I assume yet again it's just so that a "canon" incarnation of their character doesn't conflict with yours.
Then again, the Arena manual refers to the hero as "Talin" - though I've been told that this was just an example to encourage the player to embody their own character with their own background.
I believe, among the past Champions of Cyrodiil Ocato refers to, the Eternal Champion is one of them.