It's "Everest" or "Mount Everest", not "
the Everest".
"first man to climb Everest" and "first man to have climbed Everest"
are both correct grammatically, and they both mean the same
thing. Speakers will differ as to whether they believe there is
some slight difference in meaning. "to have climbed" has slightly
more emphasis on the accomplishment, even the pride of such an
accomplishment, in my opinion. But the difference is nearly
insignificant.
"first man to climb" has the advantage of being less wordy, more
direct. I would personally use "to climb" because I see no
particular reason to use the alternate form.
CJ