In a pitcher, muddle the sugar cubes with 3 ounces of the club soda until the sugar is fully broken up.
Add the remaining ingredients (except the remaining club soda) and fill the pitcher three-quarters full with ice cubes.
Stir until cold, then strain into a punch bowl over 1 large block of ice.
Top with the remaining 4½ ounces of club soda.
Garnish with the apple slices and serve with a ladle and punch glasses.