Combine sugar, mint, ginger, and 1 cup water in a small saucepan. Bring to a boil over high heat, stirring until sugar is dissolved. Let cool completely. Strain through a fine-mesh sieve.
Stir together mint syrup and guava juice in a pitcher; chill. Fill glasses with ice and 1/3 cup guava mixture. Top with seltzer. Add mint sprigs and a shot of liquor to each. Stir once. Serve immediately.
