Chop the unpeeled cucumber into chunks and puree with a hand blender. Strain through a sieve, reserving the juice.
Cut the limes into chunks and put into two glasses. Add the sugar to the glasses and muddle with the end of a rolling pin or spoon to extract the lime juice. Pour in about a shot of the cucumber juice and add a shot of cachaca. Top with ice and enjoy. You will have leftover cucumber juice for more cocktails!