Dovahkiin beats Courier 6. Dovahkiin could paralyze Courier 6 with ease, stop time, and get right up next to him with a heavily poisoned daedric sword and finish him off. The only chance the courier has is armor piercing ammo, in which Dovahkiin could just use stone flesh.
Dovahkiin can't stop time, that's the Champion of Cyrodiil.
Dovahkiin would never be able to CATCH Courier Six. Turbo + Travel Light and he simply runs faster. Toss Rushing Water into the equation and he can fire multiple times a second. Any shout simply wouldn't work because the Courier can "slow time" (slow his perception of time) far better than the Dovahkiin can. Dovahkiin can slow time, but he also slows himself in the process; the Courier is only slowing his perception of time, thus speeding himself up drastically.
And therein lies the problem: one has a gun, the other has a knife, the guy with the gun runs faster than the guy with the knife. Sure, he's got a bow too, but arrows aren't exactly the fastest projectiles and are thus easily dodged when your opponent has time moving half as fast. Just keep kiting the Dovahkiin and firing shots that deal 200 damage, 300 on a crit. Lay down Satchel charges for another 250 damage, keeping him from getting close. Hell, even if he finally DID catch the Courier, one could assume the Courier would definitely get the first few shots in thanks to Turbo and VATS. All it takes is ONE Paralyzing palm and the Dovahkiin is down for the count. Paralyzing palm + a melee or unarmed weapon that pierces defense rating and the end.
These sorts of fights are best determined by who has the most tools available to them. Courier Six can slow time (fully, not that half-assed version Dovahkiin gets), increase his attack rate to godlike speed, get maxed defense, cure poison, go invisible, see invisible targets, run fast as hell, deal upwards of 200 damage per shot, and the main factor: he has range. Range that moves fast, not slow range like a bow.