What different kinds of work did common people have or do in 1550s to 1590s Japan?
Well unlike many other parts of the world, Japanese peasants weren't just primarily farmers. While many people did grow the standard crop of the day (rice), many others practiced dry field agriculture and grew things like millets or barely. Many Japanese peasants did side things, like brew sake and make miso or lamp oil. Coastal villages produced salt. Some people became cart drivers, or worked in saw mills, or made silk.
Mining was another thing that was done all over Japan. Iron was mined but so was gold and silver, in quite large quantities. The era that we're discussing now is the Sengoku Period, Silver production increased in the period following the Sengoku era. However, it started to increase during it.
And let's not forget soldiering. The 1550's to the 1590's were a major period of war in Japan and were marked by a new kind of soldier: the Ashigaru. Ashigaru were ordinary men brought in to fight for their lord, or mercenaries.
Source: Japan Emerging, Edited by Karl Friday
(Might I add; the Edo Period is even more interesting if you're interested in economics.)