Transfer cabbage to a large bowl and separate leaves, then toss with kosher salt.
Let stand at room temperature, stirring occasionally, 1 hour.
Stir together umeboshi plums, mirin, and rice vinegar in a small bowl.
Rinse cabbage with cold water in a colander and drain.
Firmly squeeze 1 handful cabbage several times to remove excess water and transfer to cleaned bowl. Repeat with remaining cabbage. Stir in umeboshi mixture, then chill, covered, stirring once or twice, at least 1 hour.
