+1 here. Take my comments with a grain of salt since I've never completed a thru or a full marathon (yet). For now, I usually hike in 10 day blocks and average pretty high miles. I also run about 30 miles per week and run half marathons, 10ks, and the like. My first full marathon will be in the spring.
So, disclaimer aside, what I can say from personal experience is that running can't fully prepare you for hiking nor can hiking fully prepare you for running. As Oz and others pointed out it's a different emphasis on the muscles. When I hit the trail while in marathon training (speedwork, long runs, etc.), I'm usually much more gassed on the hills than I was when I was doing more cross-training on the stairmill or setting the treadmill to it's highest elevation. That has been a rude awakening. However, the cardio conditioning that fast running or moderately fast hiking share is complimentary so it's a net even in my opinion.
Conversely, after I return from a long hike (say 175 miles) it takes me about a week to recover to where I can run again. But once I have, I have much more power on the upside of the hills with slightly slower speed in the flats (a week off running isn't enough to slow you down that much, a thru on the other hand...). As others have posted, my feet hurt, my knees are sometimes sore and there's an, um, "hitch" to my step Oh, and right on with Jeff earlier about the shoe size. I've gone up a full size in my running shoes from a 10/10.5 to 11/11.5 And I need AAA width. It probably looks like a clown came through after I've hiked a muddy section.
So I guess it comes down to whether you will be running for time or just running to finish. I would certainly say that a thru hike would leave you with a good to great cardio capacity, lower overall weight and overall stronger legs, all positives in the running book. So you could, as others have done, hop right off and run a marathon. Not sure it would be your PR or feel the best but I'd definitely say you'd have an advantage on a hilly course.