The sentence is a bit informal and context dependent, not something you're likely to see in a formal setting but certainly acceptable in a colloquial sense.
"I know him from before" is missing the information about the past that would normally follow the word "before". The speaker is leaving out information and assuming the listener knows what they mean by "before", e.g. "I know him from before the war" if the listener is assumed to know that "before" must mean "before the war". Many different languages do this sort of shortening or elision where the speaker leaves out information assuming the listener will understand from context. It allows us to speak more briefly, more succinctly.
In that sense "before" isn't the noun, but the beginning of a partially elided noun phrase such as "before the war" from my example. It's clear that "before" can't function as a noun on its own: "the before" doesn't sound quite right, though perhaps you could take some creative license and say "the before-time" to force "before" into being a noun.
If the listener knows what you're talking about, a single word like "before" can take on great significance, conveying that you're referring to a time prior to some major event without explicitly referring to the event.