Socrates made the examined life a condition of the life worth living, and he arguably died for it. The Stoics built an entire philosophical practice around self-examination: Marcus Aurelius wrote the Meditations as a private exercise in catching himself failing to live by his own principles, and he did this while running the Roman Empire, which suggests he didn't find the two activities incompatible. Augustine's Confessions, written around 400 AD, offer a sustained and searching account of his own interior life that predates Freud by about fifteen centuries, give or take.
In Chinese philosophy, Mencius describes the concept of introspection as "seeking the lost heart," the recovery of something innate that gets buried under the noise of ordinary life. Shakespeare's Hamlet is a play about what happens when you're constitutionally unable to stop examining yourself and start acting, and the fact that Elizabethan audiences immediately recognized this as a problem implies they were already somewhat familiar with the practice being satirized; you can't parody a concept your audience has never encountered.