Put4 cups of water and the sugar into a saucepan. Bring to a rolling boil uncovered for 10 minutes.
Add the white vinegar and boil uncovered for 15 more minutes, until the liquid has reduced to a couple of cups. (Now you have vinegar simple syrup)
Stir in the mint leaves and boil for a minute more. Remove from heat and let cool for 30 minutes
Strain out the mint leaves and pour the vinegar simple syrup into a jar. Chill until ready to make soda.
Fill a glass with ice cubes and several pieces of cucumber. Add the syrup and seltzer water in about a 1:3 ratio, syrup to water, or whatever work for you to achieve the level of intensity you for your soba. Stir with a spoon