When will the Milky Way "arrive" at the Great Attractor, and what all happen then? The Great Attractor is described as a location to towards which the Milky Way, along with all other galaxies in the Laniakea Supercluster are moving. How long will it take for the Milky Way to "arrive" there, and what will happen when it does — what, in fact does "arrive" mean in this context?
I don't want to make any assumptions here regarding the Milky Way's presence in the Laniakea Supercluster simply because of how recent the discovery is. The findings could very well be accurate, but I don't want to base this answer off of them. Fortunately, I've found a few papers that get us around that little issue, as well as the University of Hawaii's site, that led me to them . Coincidentally, one paper bears the name of one of the researchers who recently announced the Laniakea findings, R. Brent Tully, who was also involved with the other.

The two papers (here and here) suggest that, no matter whether the Virgo Supercluster is or is not part of the Laniakea Supercluster, the Local Group (and by extension the Milky Way) is not being pulled toward the Great Attractor. Instead, it is headed for other galaxy superclusters beyond the Great Attractor, including the Shapely Supercluster. 

The idea that the Milky Way is not moving directly towards the Great Attractor could render the question moot, except that it simply shifts the destination: What will happen when the Local Group reaches the further superclusters?

Well, we don't [yet] know. As the papers say, there are a number of sources "pulling" on the Local Group, and so we can't say for sure just where the Local Group will wind up. This means that we also can't set a definitive timetable for the arrival. Finally, modeling galaxies with precision is not easy. Some theorists have used models in general relativity that treat many galaxies as grains of dust, so getting more specific is tough! We won't know just what will happen during after the "arrival" until we get a lot more data. Even the composition of the Great Attractor itself is rather mysterious.

I'll end with that odd term, "arrival". As the article you referenced (which, unfortunately, misses a lot of the better points of the findings, such as actually describing Laniakea) says, 


  The giant structures making up the universe often have unclear boundaries.


That's true on a smaller scale, too (see Where does the Milky Way end? for an interesting discussion). We do know which galaxies are in the Local Group, but not where the Group's gravitational influence ends and other galaxy groups begin to dominate. We could define "arriving", though, at the instant that the destination supercluster beings to attract one of the galaxies in the Local group that are further out, instead of the now-probably-merged Milky Way and Andromeda galaxies. At that point, the Local Group could be distorted by the gravity of the galaxies in the supercluster, or could simply merge into the supercluster without any problems. But like I said earlier, it's too early to do anything but speculate.