Dig with a spade shovel on both sides of the culvert pipe to remove any mounds of mud and debris that may be obstructing the culvert pipe. Obtain your neighbor's permission if the pipe runs onto his property.
Spray the center of the pipe with a pressure washer to remove as much debris as possible. Set the pressure on the highest setting possible. This will remove some of the mud and debris, and at the very least it will soften the dirt so you can dig it out.
Dig away the remaining mud and debris from the culvert pipe using the spade shovel and a long-handled hoe.
Spray out the culvert once more with the pressure washer until you can see the sides of the pipe.
Move any dirt that may have fallen into the ditch surrounding the culvert. Ensure that the culvert pipe base is in-line with the ditch bottom.