So Riven will be captured and taken back to Noxus in the future

The Awaken cinematic is not confirmed canon (yet).

how can a bad person like riven make noxus better?

It's possible, but not guaranteed. As Riot cottonfxn said:

For the purposes of this video's storytelling everything is just slightly in the future. It portrays a plausible, potentially real future.

Then tyler one will tryvto execute her in the coloseum.

Interesting. Although that feels like an absolute ton of chapters/stories would be missing between Riven now and Riven there.

She's in Ionia up untill a supposed new war with Noxus begins. Riven has to either get involved or simply be spotted, which neither of would occur instantaneously. Perhaps a few battles, skirmishes won and lost, perhaps she goes into hiding or seeks aid. Eventually, someway and somehow she's recaptured, but not executed on the spot for desertion? Then for something more unfathomable, they ferried her back to the capital to deliver the punishment, what and why?

If this is even vaguely how it all transpires, it would take a few years of stories to release and bridge the gap. That is unless the devs skim over a lot of detail and build-up just to make the cinematic "canon" within the year.

Im 90% sure the awake cinematic is not canon also I could be wrong but I can't remember anything she did against noxus other then leaving and noxus does not seem like a place where they would care about that.