Cut cucumber crosswise into three equal pieces. Peel and chop 2 pieces, then put into a blender. Peel 4 lengthwise strips from remaining piece, leaving stripes of green, then halve piece lengthwise and core. Thinly slice crosswise.
Add water, sugar, fresh ginger, and lemon juice to blender and purée until smooth. Pour through a medium-mesh sieve into a pitcher, then stir in sake, crystallized ginger if using, and cucumber slices. Chill, covered, until ready to serve, up to 2 hours. Serve over ice.
