In a small pot over medium heat, combine the sugar, water and the ginger. Bring to a boil then shut off the heat and let cool to room temperature.
Fill 4 glasses with ice. Pour seltzer water into the bottom third of each glass. Add 1/4 cup of the ginger simple syrup to each glass and gently stir. Put a teaspoon upside down over the glass and pour 1 shot of rum over the spoon, so that the rum stays separated from the seltzer.
Put slice of ginger and lime wedge onto each skewer. Garnish each drink with the skewer and enjoy.