For the Cucumber Shot Glasses: Cut 4 (2-inch) long pieces out of the cucumber. Use a melon baller to carefully scoop out flesh from 1 end, leaving a 1/2-inch bottom on the other end. Repeat with remaining cucumber pieces, reserving cucumber flesh.
For the Sake Shots: Combine cucumber flesh with lime juice in a blender. Blend until smooth, about 15 seconds. Pour cucumber puree through a fine mesh strainer, reserving 1/2 cup of cucumber juice. Add sugar and set aside.
Pour watermelon rum and sake into a cocktail shaker filled with ice cubes. Add reserved cucumber juice and shake vigorously. Pour into cucumber shot glasses. Serve immediately.