Again, this is a matter of perspective. The amazing long lived Earth observer would see the universe expand out of view, and you along with it wouldn't they?
Yes, after billions of years you would move outside of the sight horizon of the long-lived observer on earth and disappear from view. For you, the traveler, this would happen in mere minutes. But you wouldn't have crossed the universe in that time because the edge of the visible universe is constantly expanding away from us faster than we can travel to catch up with it. Even if we travel at the speed of light.
Lawrence Krauss has given a talk that mentions something similar. He says that we live in a good time because we can still see "everything" around us. At some point in the far future, any observers won't be able to determine many things about the universe because the "stuff" in it will be too far away to observe.
Even wilder to me is that one day, anyone living in the Milky Way will look up at the sky and be unable to see any other galaxies. If it weren’t for archaeological evidence they would never have any way of knowing about the existence of those other galaxies.
