I wouldn't be surprised if some people get to the center in less than a week. There's a somewhat consistent tradition of players finishing games in the first few days after release, way faster than what the devs expected, so this one wouldn't be that much different, but of course it depends on just how restricting the gameplay mechanics are, especially hyperjumps and refueling, not to mention other artificial barriers like needing better shields or armor to survive the harsh space near the center, which would force you to make longer stops, mine for resources and buy better equipment and ships. The trip to the galactic center in Elite: Dangerous took a little over a week.
As for finding other people in the game, I think players will pull it off much faster than people think, but again, it depends on how they handle the galactic map. Murray's mentioned that there'll be a sort of a "galactic north", a little mini map in the corner of the fullscreen galactic map showing your orientation etc. Having coordinates would help immensely, but even without them, people can take videos and screenshots of their locations in relation to the grid/compass/whatever, and combined with the name of said solar system people will at least start to flock to certain clusters, most probably near the center, or at least the inner half of the galactic radius.
It'll still be tricky, but far from impossible or lasting a lifetime. And once people are in the same solar system, they'll arrange a meeting at a specific planet and at a specific beacon on said planet.
So, if we're talking about playing the game "normally", as in not caring much about finding other people, then the chances of finding them would be slim to none, probably higher once you're very close to the center. But if we're talking about an organized effort, with a community focused on nothing else but finding other people, then the whole thing becomes much more probable.