Combine the sugar and 1 cup water in a small pot and bring to a simmer. Bubble gently for 1 to 2 minutes. Remove from heat, and cool completely.
To make the shaved ice, squeeze the limes and lemons into the simple syrup, discarding the spent rinds. Add the cucumber chunks, and puree the mixture with either an immersion blender, or in the container of a regular blender.
Using a mesh strainer, strain out all of the cucumber pulp, and any seeds. Chill the syrup until you are ready to use.
Spoon the shaved ice into 2 large glasses. Pour half of the syrup over each serving of shaved ice. Serve immediately garnished with cucumber, lemon or lime slices.