I would definitely cycle your 5 gallon, the added stability and less upkeep it takes is so worth it, even if you have to buy a filter.
I would consider cycling with your fish in it though. You would need to do daily water changes of 25% and a larger one every couple of days, but if you keep up with the water changes your fish will be fine and have no stress. Think about it, cycling with a fish is in all reality pretty close to just keeping it in an uncycled tank. While you don't want to do 100% water change while cycling, most of the bacteria lives on the rocks, filter, decorations, exc. not in the water so you can do large water changes without affecting the beneficial bacteria.