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, in the week Saitama has to log a heroic act or be struck off the C-class register. Not a motivational line in context: he says it on his way home from not doing it.
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 and the wording is the dub's. What the motivational accounts get wrong is what it means.
The verdict
True. CBR's ranked list of Saitama's lines gives the episode and prints the sentence as it is spoken. Wikiquote has no One-Punch Man page at all, so this is a case where the editorial tier carries the whole weight, but the episode is easy to check for anyone with a streaming subscription: it is the one where the Hero Association's rule that a hero must log activity every week or be removed from the register comes due, and Saitama, newly registered at the bottom of C-class, has done nothing all week.
What the scene actually is
He is walking home. He has spent the week failing to find a crime, been mistaken for a menace, and decided that the problem of being struck off is one he can face later. The line is not a philosophy of living in the present; it is procrastination, said by a man who can end any fight in a single punch and cannot be bothered to look for one. The episode is a comedy about bureaucracy: the strongest being alive is about to lose his licence for paperwork reasons, and his response is to go to bed. That is the joke, and the joke is the point.
Why it became a poster
Because stripped of its scene it reads as stoic calm, and Saitama's blank face on the accompanying image supports that reading. The line has been repackaged as advice about not worrying, on wallpapers and clothing, by people who have not seen episode six or have forgotten it. It is a real quote used to mean something the show is making fun of, which is a different failure from fabrication and a common one. Compare the Vegeta gym slogan, which was never said at all, with this, which was said and meant as a gag.
What makes it a good line anyway
The series is about a man whose strength has emptied his life of stakes, and "tomorrow's me" is that condition stated as a shrug. Nothing can hurt him, so nothing is urgent, so every problem can be deferred. Read that way the sentence is sadder and funnier than the poster version, and much more the show's. The people quoting it as motivation have got hold of a real thing and pointed it the wrong way.
How we checked
The verification process requires a primary or editorial source; here an editorial ranking gives the episode, and the scene is where it says it is. 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, with the note that the meme has the words right and the meaning wrong.