Yes. Leave ASM up and running but shut down the DB. The upgrade option shuts down the ASM instance as part of the upgrade process. And do not touch the oraInventory. Oracle upgrade processes takes care of the necessary updates to the inventory.
even though I see this answered, I do recommend that you move your oracle inventory out of you previous OH, you should put it in a location that should be independent of your OH as with upgrades you will normally at one point in time remove your previous ORACLE_HOME