Boil one cup of water. Steep green tea bags for 5 minutes. Remove bags and stir in honey. Allow to cool for about 10 minutes.
Fill two glasses with ice.
In a cocktail shaker, muddle apricot jam, mint leaves, and lime juice. Add 1/2 cup brewed green tea and the rum. Add ice and shake until cold. Strain evenly into both glasses and top with sparkling water. Garnish with fresh mint leaves.