He does! Sunesu's favorite hobby is climbing, and he is pretty much always ready to stretch out and scamper up something. It's both a useful skill to train and (usually) a chance for some solitude, at least if he climbs high enough.
His other hobbies include collecting knick-knacks and trinkets from his journeys, especially mammets (his small house is slowly running out of space), and fishing. Being a fisher from a young age it actually took Sunesu a while to wrap his head around the concept of fishing for sport and leisure rather than for sustenance, but the guildmasters at the Limsa Lominsa Fisher's Guild got through to him eventually and converted him.