I think it's a little too much of a stretch to tie the "more than 300 years ago" time frame to Voyager's trip through the black hole. The dialogue for the sequence that reveal's Vger's identity goes as follows:
KIRK (rubbing at the nameplate): V...G...E...R. Vger.
He rubs again at the nameplate.
KIRK: V...O...Y...A...G...E...R. Voyager! Voyager 6!
DECKER: NASA. National Aeronautics and Space Administration. Jim, this was launched more than 300 years ago.
KIRK: Voyager series. Designed to collect data and transmit it back to Earth.
DECKER: Captain, Voyager 6 disappeared into what they used to call a black hole.
KIRK: It must have emerged on the far side of the galaxy and fell into the machine planet's gravitational field.
From that point Kirk, Spock, and Decker start the discussion of what happened to it after the machine planet discovered it. After Decker's original notation of Voyager's launch, the 300 year figure is never mentioned again.