Having killed literally thousands of farm geese (research farm in Edmonton - don't ask...), I can say that there is no way you'll be holding it for very long with one arm if you break the neck up high. The flapping is unreal when ethics-approved cervical dislocation is done, but nerve traces show the least pain reaches the animal's brain when done that way.
For field solutions, taking a knife and cutting the throat works; the best solution, which deals the least pain during the "coup de grâce" without using a firearm, is to carry a pair of wire nippers and cut/crush/clip the spinal cord up high from the rear. This may lead to some flapping and breast tearing, so I'd recommend trapping the animal (wings folded tight to the body, like it would be at rest) under your groin/knees. Think "goose rodeo."
If not hunting with a dog, the animal will likely be in open field and not piled next to you and your hunting partners (if any) - by far my favorite finishing method in this circumstance is one barrel worth of shot right to the head, like one would breach a door. Up close, barrel contact, no chance of failure.
Manual cervical dislocation without tools is difficult on geese - their necks are surprisingly strong. This coming from someone who has done over 6,000 chickens with bare hands and knows the tricks of the trade. The flap, if held with just your arms, is unreal - we used to use a plywood box with a hasp, and an emasculatome to crush the necks. The box, set on a bench or chair, would bounce from the violent body action of the goose when the spine was crushed and the cord severed high near the brain stem.
Getting a goose wing in the groin will wind you like a front kick from a women's libber who thinks you invented the bra... I don't recommend it.
Short answer: give it a shot with the gun, directly to the head. Failing that, cut the throat or break the neck in a field expedient way with a multi-tool or nippers.
-M
