No, I don’t believe the average Shire-hobbit was aware of the Elves’ new year. All the evidence points to them knowing or caring very little about places or peoples beyond their borders. But Bilbo was an exception, as Tolkien made abundantly clear. How much he knew about Elves and the Reckoning of Rivendell can, I think, be inferred from descriptions of his Tookishness. As a hobbit-lad, he pestered Gandalf with questions about Hobbits who had “gone off” and he absorbed the wizard’s tales about the wide world, which surely involved Elves and enhanced Bilbo’s interest in them; as he grew older, he gleaned information from Dwarves and other travelers he met on his days-long wanderings (per Gandalf, Unfinished Tales). From this I infer Bilbo knew enough to accurately mark yestarë on his personal calendar and to have departed Hobbiton for the occasion.

Perhaps I should have used a modifier to specify the glaring omission is apparent to me. This is how I’ve always understood Bilbo’s knowledge of Elves before ever visiting Rivendell.