10 hours of prep time. The whole world is their battlefield.
Cartman should win as long as psychological warfare is used. Stewie is a tech genius but it's been shown he cant handle certain things such as the rape of Rupert, seeing Meg do "things", Olivia's treatment of him, and I think he's got into a certain spats with Brian that left him mad. Also Stewie is an on and off fighter. Remember Lethal Weapon 4 and the Death of Louis but he got destroyed by Joe's kid. Cartman can fight believe it or not as long as he uses Sumo.
Lol Yea dude, Stewie versus Dexter (Lab) Would be a much better fight tech-wise. But since this is simply preperation over-all, I can see Cartman winning pretty damn easy. Cartman is a prep God in his universe, Stewie who is very intelligent in His own right, is still a baby. Cartman will know that Stewie is attached to Rupert, as Daak pointed out. I can see Stewie coming for Cartman full force, with a Jet-pack and Lazer Gun, Rupert by His side. I can see Cartman some how getting his hands on Rupert, or causing Stewie to damage Rupert somehow, mentally breaking Stewie down. With that second that Stewie is shocked, Cartman kill's Him. But h2h Stewie destroys Him, also with gun's and melee weapon's as well. Cartman wins in this situation.
With 10 hours of prep Stewie could go back in time, build a doomsday device, pull out whatever tech he has in his closet, or simply bring a firearm to the battle.
This might as well be "Random street leveler vs Reed Richards with prep."