Did Saitama Really Say “I'll Leave Tomorrow's Problems to Tomorrow's Me”?
Yes. It is from One-Punch Man season one, episode six, the episode in which Saitama learns that C-class heroes are struck off after a week of inactivity. CBR prints the line; the exact scene is not sourced.
Yes. Saitama says "I'll leave tomorrow's problems to tomorrow's me" in the first season of One-Punch Man, episode six, "The Terrifying City." The line is real. What no source we can read establishes is the moment in the episode where he says it.
The verdict
True. CBR's ranked list of Saitama's lines gives the episode and prints the sentence in its body text, not only in a heading. Wikiquote has no One-Punch Man page at all, so this is a case where the editorial tier carries the whole weight. CBR does not say whether its wording follows the subtitles or the English dub, so treat the phrasing as one English rendering of the line.
What happens in the episode
Episode six is the one in which Saitama, newly registered as a C-class hero, learns that C-class heroes are dropped from the register after a week of inactivity and rushes off to find a villain. Over the episode he refuses Sonic's demand to settle their rivalry, knocks Sonic out when the ninja starts causing mayhem, and kills the monster Kombu Infinity. None of the sources we can check places the line among those events, so this page does not claim a scene for it.
Why it became a poster
Stripped of any scene, the sentence reads as calm advice about not worrying, and that is how it gets repackaged. CBR reads it the same way, as the essence of a man who does not let himself be consumed by worry unless the matter is pressing. That reading fits the premise: Saitama is a hero who has grown bored because every fight ends in a single punch, and "tomorrow's me" is that condition stated as a shrug. Compare the Vegeta gym slogan, which was never said at all, with this, which was said.
How we checked
The verification process requires a primary or editorial source; here an editorial ranking gives the episode and prints the line in its body text, and Wikipedia's episode summary gives the episode's events. One-Punch Man does not yet have a hub on this site, so the line is not in the quote corpus, which requires two sourced lines per character before a page exists. It sits in the fact-check database as true.