Tajik: Not exactly. He didn’t exactly promise Olam Haba, and it’s not specific to Rosh Hashana. Here’s the main quote from Sichos Haran 141:
“Bear witness to my words. When my days are over and I leave this world, I will still intercede for anyone who comes to my grave, says these Ten Psalms and gives a penny to charity. No matter how great his sins, I will do everything in my power, spanning the length and breadth of creation , to save him and cleanse him….”
In addition, in Chayei Moharan (a collection of various oral traditions about Rebbe Nachman from Rabbi Nosson of Breslov), the following sentence is added: “I will pull him by his peyos out of gehenom.”
A little unusual? Yes. But the Zohar says a tzaddik is greater after death than while living, so it’s conceivable that a tzaddik could “intercede” in some way with one’s soul after death. And I think it’s obvious that the part about the peyos is meant to be figurative, not literal. Obviously, peyos don’t come with you to the afterlife. It may sound like he’s promising Olam Haba, but I don’t think that’s the intended meaning.