Stir ice cream and matcha together in a bowl. Freeze until firm, 3 to 4 hours.
Combine 1/2 cup sugar, the sake, ginger, and 1/3 cup water in a medium saucepan over medium-low heat. Cook, stirring, until sugar dissolves. Add plums and simmer over medium heat, covered, until fruit begins to soften, about 5 minutes.
Transfer plums with a slotted spoon to a bowl. Discard skins and ginger. Return liquid to stove; boil until reduced to 1/2 cup, 7 to 10 minutes. Pour over plums and chill until cool, about 20 minutes.
Whisk cream with remaining sugar in a bowl. Fold in candied ginger.
Spoon a third of plums and sesame snaps into 4 sundae glasses. Top with a scoop of ice cream. Repeat layering and top with remaining plums. Dollop with whipped cream and sprinkle with remaining sesame snaps.
*Find matcha at well-stocked grocery stores and at breakawaymatcha.com; find fruity nigori sake at liquor stores.