Do you have access from both ends? What's the diameter? If it's a 16" pipe, just to choose a size at random, then it's got about 14 cu. ft. of dirt in it, say 1/2 a yard. That might weigh between 1,000 and 1,200 lbs depending on the nature of the soil and how saturated it is. I think 60 acre hillside's solution is very clever, but it depends on being able to move all the dirt at one time. I'd worry that there's a high probability of pulling the dirt into a plug within a few feet, and damaging the culvert. Be careful if you try that.
A much more difficult way to remove the dirt, but without the danger of doing damage, would be to attach something to the handle of a hoe so that you could eventually get it 10' into the culvert, and start dragging the dirt out from both ends. You could try the tractor method regularly until there was enough dirt out of the way for it to work.