Another method of sanding a neck cork is to get a wooden dowel close to the inner diameter of the neck tenon and then sand or build up the end with masking tape to fit very snugly inside the neck tenon. This dowel is then held vertically in the vice.
A variation of this idea that I like to use is to find a wooden dowel that fits snugly inside the mouthpiece end of the neck. You then drill a hole that diameter in the front edge of your workbench. Next the dowel is cut to the length that allows it to go completely into the drilled hole with about 1" extending. This dowel extension is then placed inside the small end of the neck, and the back of the neck is supported by the abdomen of the tech seated at the workbench. Those familiar with the use of a bench "end peg" will recognize the similarity to the method described. The advantage of the bench dowel/abdomen holding method to sand the cork is that the neck can be turned 360 degrees to sand evenly on all sides.
The actual sanding is done using long strips of emery cloth or sandpaper about 1/2" wide with a "ragging" or "shoeshine" motion. If strips of emery cloth are not available you can add duct tape to the back of a sheet of regular sandpaper and then cut it into strips. I like to do an equal number of back and forth strokes from the North, South, East, and West to keep the cork as round as possible. I prefer 120 grit to start, and when the cork gets close to the correct size, I finish it with 800 - 1000 grit to leave a smooth, professional finish.
It is important to sand the cork to the same diameter throughout its entire length to create a cylinder rather than a truncated cone or "frustum". I like to fit the cork so that the mouthpiece will go completely to the end. Once the cork is smoothed with the fine sandpaper, I melt paraffin wax into the surface and then add a generous amount of cork grease for the initial mouthpiece insertion. I put the mouthpiece on slowly using a back and forth twisting motion till it gets to the end of the cork. Then I will use the edge of the mouthpiece to trim the end of the cork to make a perfect circle.
This process takes more time and effort, but in my experience the cork remains functional for years.