To be "on the natch" is to be in a natural state, without being intoxicated by drink or drugs. "Natch" being a shortened version of natural and "On the" being a prepositional prefix.
In context, to be on the natch would be going on a night out sober.
"Hey you out tonight?"
"Yeah, I mean I've got no money so I'll be on the natch tonight, but I'll hang out with you guys."
"No one has ever enjoyed your company sober, you are a depressive, drag on all our buzzes and we all wish you would die."