Drunken ring is good for any build. But you can also go without it on any build.
Basically - 1 stance ring is a must (you`ll not get +1 AR from Promotor without it), 1 drunken ring allows you to get +1 AR from Drunken instantly, and get +2 AR from it a bit earlier.
If Barrol die often - you`ll need Drunken ring to DPS more.
If you want to clean missions as fast as possible - you`ll need ring to get +1 AR instantly.
If you`re playing casually and do not mind 30-40 more seconds to your mission runs - you can skip Drunken ring and go with 1 Prid.