In a small saucepan, combine the ginger, sugar and 1 cup water over medium-high heat. Let boil, and then reduce to medium until the liquid becomes syrupy, 15 to 20 minutes. Remove, strain and cool. Using a martini shaker to mix each drink, add ice, 1 ounce rum, 1 tablespoon ginger syrup, the juice of 1 lime quarter and 1 tablespoon apple juice. Shake well, pour into a rock glass, top with 1/2 cup beer and garnish with a thin slice of apple and freshly grated ginger.