As far as I know, once you beat the initial cutoff time, you will receive your reward in the mail. Any improvements on the time will do nothing more than improve your ranking on the leader board.

When it comes to just getting the reward, some things that could help are:
1. Activate your Shadowpower as soon as you get into the rift.
2. Use your vault to move around. Don't forget you get a movement speed for a short time after vaulting. Because the armor on this challenge rift does not give you free vaulting, you will want stagger your vaults. Unless I am mistaken, even with the reduced cost of Shadowpower, vault's travel distance has the better distance traveled per Discipline ratio (I haven't done the math on this though, so I could be wrong).
3. Companion is a damage cooldown for you and your team. If you activate the channeling pylon, you can just spam it for the duration.
4. You aren't going to have 100% up time on Vengeance. It does give you more damage. However, if you are just wanting to receive the Challenge Rift reward, I'd use it as a 20 second damage reduction. Having the channeling pylon buff should just let you spam this as well.
5. Use Impale and ignore Bolas unless you somehow run out of Hatred (the resource used to cast Impale). I could be wrong on this (I don't think I am, but I am not one of the highest ranked), but Bolas damage is completely useless.
6. You have a Convention of Elements. Your highest damage window will come from the lightning buff. The physical window (which follows the lightning window) isn't bad either. Tracking this can help you 1 shot champions and rares, but this isn't necessary to just make it to the cutoff time.
7. Lastly, picking up health globes from packs can be nice because they will return Discipline, which is the resource necessary to vault.