Possibly not literally true, but certainly true in spirit. That's what the biography claims -- that is, the biography by Kritovoulos that Mehmed commissioned. Kritovoulos doesn't specifically tell us that Mehmed visited the tomb, but Mehmed is reported to have specifically enquired about the tombs when he visited Troy. It seems he knew Homer's reputation, though it doesn't seem likely that he read Homer himself.

Here's Kritovoulos' text, attached to the year 1463 (Kritovoulos 4.11, pp. 444-445 Müller):

And when he arrived at Ilion he looked at its remains and the traces of the ancient city of Troy, and its size, its terrain, and other advantages of the site, and how it lay in a significant location for the land and sea. He went on to enquire about the tombs of the heroes, I mean Achilleus and Aias and the others, and he praised and congratulated them for their fame and their deeds, and because they met the poet Homer.

He is reported to have said, even shaking his head a little: "God dispensed to me the job of avenging this city and its inhabitants after so many ages. For I subdued their enemies and sacked their cities, and I have made them the spoils of the Mysians. For the people who sacked this city of old were were Hellenes, Macedonians, Thessalians, and Peloponnesians: their descendents, after so many ages, have paid the penalty to me now, for their hybris to us Asians, both at that time and frequently in later times."

(The bit about meeting Homer, incidentally, is down to ancient biographical traditions -- in some variants of Homer's biography he was supposed to have been an eyewitness to the Trojan War.)

Mehmed wasn't the first to claim the role of avenger. Xerxes did, too, when he passed Ilion on the way to his campaign in Greece in 480 BCE, supposedly saying that he was avenging king Priam. I doubt he literally believed that, but it was damn good PR.

So, visit to Troy, yes; visit to Achilleus' tomb, maybe (but not only Achilleus); offerings to the hero, no.

It's not clear if there was still an identifiable tomb -- or rather, burial mound -- associated with Achilleus and Aias in 1463. In earlier times, at least, there were mounds that were supposedly those of one or other of the heroes. When Xerxes visited, according to Herodotos, he offered sacrifices to Ilian Athena and libations to the heroes. When Alexander visited in 334 BCE, according to Arrian and Plutarch, he and his sidekick Hephaistion reportedly made ceremonial visits to the tombs of Achilleus and Patroklos respectively -- running naked to the tomb and laying a wreath. Dion Cassius tells us that in 214 CE the emperor Caracalla visited and dedicated a statue to Achilleus. There's a report of a visit by emperor Julian to Achilleus' tomb, too, though I can't track down the primary source for that just now.

[Here]s a 2004 article](https://www.jstor.org/stable/25067103) that frames Mehmed's visit in the context of the long-standing tradition of understanding various conflicts over the millennia as east-vs-west. Jonathan Burgess has more details about Achilleus' burial mound(s) in this 2005 article on imagined burial mounds of Achilleus -- look to the section "Political Visitations" for a list of people who came to visit the mound that was supposedly that of Achilleus. Burgess goes on to talk about ancient traditions and more recent attempts to "identify" Achilleus' mound. Troy was a moderately important city until a major earthquake in late antiquity -- rulers regularly gave the city special privileges from Alexander onwards -- but it's doubtful if there was still anyone living there by the time of Mehmed's visit. So it's equally doubtful whether there were still any locals to pass on traditions about the tomb to Mehmed.