Cook the noodles in boiling water until tender. Drain and rinse in cold water.
Drain thoroughly and transfer to a medium-sized bowl. Add vinegar, sugar,
soy sauce, and salt. Mix well.
Cover and chill until cold.
Peel and seed the cucumber. Cut into quarters lengthwise, then cut into thin pieces.
If not serving right away, wrap the cucumber pieces in a plastic bag or plastic
wrap and refrigerate.
To serve, divide noodles among 4 serving bowls. Top with a handful of cucumber slices and, if desired, very thin slices of scallion greens and sesame seeds.