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If I understand right, your main problem is water flowing down the hill and washing into/across the run? Your best hope of a solution would be to dig a swale (wide shallow trench) just outside the run on the side that has the hill, so that the swale is intercepting the water and directing it harmlessly away elsewhere. While a bobcat or sumpin' like that makes the job much easier and quicker (also messier), especially if the run is relatively long, this is absolutely 100% something you can do by hand with a shovel. You will probably start with a too-narrow too-shallow ditch, but it will be better than nothing, and you can continually enlarge it as time and energy permit, until it seems to be doing an adequate job. If erosion in the swale is a problem, seed it with grass during the non-floody season, and/or put down gravel or etc. to limit erosion.
Mind, this assumes there IS some 'harmlessly elsewhere' for you to direct the water TO... usually you can, but if the coop is at the lowest part of the landscape, you may be stuck with the ultimate Plan B which is to raise the level of the run up above the level of the flood. The cheapest aggregate you can buy by the truckload is usually called 'roadbase' or something like that (here, it is "A aggregate") -- it is a mix of dirt and sand and small gravel and big gravel, and is pretty cheap by the truck dumpload. Build short STRONGLY ANCHORED retaining walls and fill with roadbase, or sand or non-sharp gravel or whatever, til the run is high enough that the water pools *around* it not in it.
Good luck, have fun (er, maybe not "fun", but anyhow good luck
),
Pat
