If u wanna use it, use it on The Emperor the root does not do any significant damage on Alduin
I'd save it for big Al or some other enemy you have trouble with. You can kill the Emperor with weapons or magic so there's no need to poison him.
Can its poison 1-shot Karstag or the Ebony Warrior?
I don't think so. I'm not big on alchemy and I usually use the Jarrin Root on the Emperor so I don't have any firsthand experience with what the Jarrin can do. I think with the right ingredients, the Jarrin Root can do around 2,000-3,000 damage but that won't kill The Ebony Warrior (2071 health and 50% resistance poison due to being a Redguard) or Karstaag (4,000 health). You'd need to resto-loop or use mods if you want to one shot them with poison.
Does poison work at Alduin at all?
The root has a great role-play-aspect. At least, you can show that you are able to use some trick beside what a warrior would do.
On the other hand, casting (or using a scroll) of mayhem and let the emperor be killed by it's own guard has also something.
If you're quick about it, you can hand over the Jarrin Root to the chef, open your inventory, and then drop it before she actually takes it from you. She'll carry on as if you gave it to her, and then you can pick it up again and save it for later.
There is also the duplication glitch can be used to dupicate almost anything.
Karstaag has 100% immunity to poison ..........so no ......you will just waist the Jarrin Root
your very correct about the Ebony Warrior, due to him being a Redguard he has a auto 50% poison resistance