There's nothing "cyclic" in the universe Hawking is describing. Rather, Hawking's thought experiment applies to a particular shape of the universe.
The observable universe is, for practical purposes, a sphere 93 billion light years in diameter, with us at the centre. Observations suggest that this observable universe is pretty much isotropic and homogenous. If anything exists beyond this sphere, it's so far away that the light from it hasn't had time (i.e. the age of the universe) to reach us.
It's rational to assume that there is a universe beyond what we can see. This is called the global universe. Since we can't observe it, we can only theorise what its properties might be. There are three broad possibilities for the shape of an isotropic global universe:
flat (no curvature)
closed (positively curved)
Current observations indicate that the universe is flat, with only a 0.4% margin of error - in other words, if it's a simply-connected universe (e.g. not a torus) then it extends infinitely in all directions. Hawking's thought experiment, on the other hand, takes place in a positively curved universe. This can be thought of as a three-dimensional hypersphere. One way to picture this is as though we're an atom in the surface fabric of a balloon: from where we are, everything looks flat and infinite, but in fact the flatness curves away to form a spherical shape. Importantly, this shape is "closed" - in other words, finite, as is the surface area of a sphere.
What Hawking is suggesting is that if you set out in a straight line on the surface of this balloon, and so long as your movement away from the starting point is faster than the rate the balloon is expanding, then eventually you'll circumnavigate the balloon and end up where you began.
It's purely meant as a demonstration of what a closed (positively curved) universe "looks like". Given that our local, observable universe appears very close to flat, the curvature would have to be so small that the "balloon" would be truly immense; circumnavigating it would therefore take so long (say, a billion billion years) that our Milky Way would be unrecognisable, if it existed at all, by the time you returned to your starting point.
To be fair to Hawking, in 1988 when his ground-breaking A Brief History of Time was published, there was good reason to think that the universe was positively curved, since a truly flat universe (curvature exactly zero) seemed an improbable "special case". However, the discovery in 1998 of the accelerating expansion of the universe, which provided direct evidence of dark energy and firmly installed Lambda-CDM as the preferred model, rewrote the science and was probably one of the reasons Hawking decided to rewrite his book.
Even so, when A Briefer History was published in 2005, only the earliest data from the Wilkinson Microwave Anisotropy Probe (WMAP) was available, putting the total density of the universe at 1.02 (allowing a realistic possibility of a slight positive curvature), whereas by 2012 the WMAP data had reduced this to 1.0027 (very close to flat).