In a pitcher, combine the simple syrup, apple juice, rum, and sparkling water. To serve, fill a martini shaker with ice and add about 1 cup of the rum mixture. Shake well and pour into martini glasses. Garnish with apple slices and serve.
In a small saucepan, combine the sugar, water, and ginger over medium heat. Bring to a boil, reduce the heat, and simmer for 5 minutes, stirring occasionally, until the sugar has dissolved. Remove the pan from the heat and allow the syrup to cool, about 20 minutes. Strain before using.