It has axial symmetry, but it doesn't have cylindrical topology. (Technically sections of it could be cylindrical, but in that case the global topology must be toroidal. The diameter of the cylindrical section then must be also be much larger than twice the critical Gödel radius.) While it's easier to see the behaviours of some families of geodesics (notably those that intersect the rotational axis) in a cylindrical coordinate chart, the choice of coordinates never determines the geometry of spacetime.

We can, if we wanted, apply cylindrical coordinates to a galaxy cluster in our cosmos (technically a particle of the non-radiation perfect fluids in the FLRW solution) or on any of the particles in the Gödel solution not on the Gödel axis.

The interesting principal point made by Gödel is that we can't reasonably work with a foliation along the axis or along the geodesics traced out by any of the particles of matter, whereas even at the time of publication (1949) there had been about thirty years of doing just that with various cosmological solutions (and approximations) to the Einstein Field Equations.

This was one of the notable early shots in the "problem of time" discussion which continues. The two main features that arise from Gödel (some of which were discovered many years later, as in the work by Kerr, but were suspected some years beforehand, e.g. by Gamow) is that angular momentum in General Relativity can break the strongest notions of causality ("global hyperbolicity"), and that it is the distribution of stress-energy ("matter") that can pick out easy-to-work with frames of reference. In other words, Gödel showed that rotating massive systems might not in general work with formalisms in which time is a background parameter.

Fortunately it turns out that \Lambda-CDM admits a foliation (and thus a time parameter in the form of the scale factor): galaxy clusters in that model are essentially para-Newtonian "Euler" observers (of the vorticity-free perfect fluids, and in particular the radiation fluid), and so we can fix a set of spatial coordinates on them where they do not move against those coordinates, even as we expand (or even contract) the cosmos. These are the cosmological comoving coordinates. Galaxy clusters in a Gödel cosmology are not Eulerian at all: they twist around each other, and that frustrates the fixing of any such set of coordinates. In Gödel, every galaxy sees galaxies rotating around it, and no galaxy has an observable like the CMB multipole that is "still" for them. Again, we can do the gravitational physics of very different cosmologies in any set of coordinates we like (even arbitrarily difficult to use ones, in principle), and even in no coordinates at all, although we may find some choices lead to us missing out on the tractability and conceptual utility of 3+1/initial-value formalisms and perturbation methods.

> unlike ours

The most obvious difference is the absence of red-shifting of distant galaxies (bottom left column, p. 450 Gödel 1949 <https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.21....>) -- it's impossible to introduce this into Gödel's solution. That we have redshifting (and a strong angular-diameter/redshift/brightness relation) (and local Einstein-Maxwell) eliminates the Gödel metric as a good description of our universe. We don't have to consider tuning of the free parameters of the Gödel model.

> Showing we don't live in that universe simply because it has time travel still eludes us.

How do you arrive at this conclusion?

The critical Gödel radius, coupled to slowly-moving galaxies that form by gravitational collapse (cf. Gödel ibid. p. 450 top of right column), produces some observables that are hard to interpret as seeing emissions from a much older copy of a near-neighbour galaxy. Those copies can in common enough circumstances coincide (see e.g. fig 19 of <https://iopscience.iop.org/article/10.1088/1367-2630/15/1/01...> and following discussion). The brightness part of the angular-diameter/distance/brightness/age relation in our sky is probably impossible to reconcile with the absence of galaxies much older than M31 in our sky. Surface brightness fluctuation methods are enough to disqualify the presence of a Gödel critical radius. This would be especially true in luminous red galaxies (which would have different clustering and different large-scale correlations with the baryon acoustic peak).

Additionally, the coldest CMB cold spots do not remotely fit a Gödel future-image-riddled spectrum.

It's not that there is time travel in Gödel's universe that is interesting: it is that there is so much of it by design, since Gödel wanted to think about whether 3+1 splitting could be done on general curved spacetimes. He proved it cannot. We now know we need certain conditions such as the closure conditions (each spacelike hypersurface Sigma has a spacelike geodesic from every point to every point in the hypersurface; every maximal spacelike geodesic has a nonempty open segment in Sigma; every maximal timelike geodesic intersects Sigma) the "thin" and causal conditions (no nonempty segment of a null geodesic is found in Sigma, there is no timelike curve between any two points in Sigma) are also needed for a 3+1 approach. Gödel's universe breaks the "thin" and causal conditions (it even goes further by presenting timelike geodesics connecting points in Sigma, and maximal timelike and null geodesics can intersect Sigma more than once).

"Time travel" in Gödel's solution has a rather restricted flavour compared to sci-fi ideas of time machines (in particular in Gödel's universe you don't get to time-travel versus something you are gravitationally bound to, so no time-travel strictly within a galaxy -- you only get future images of objects well outside your galaxy), and this motivates questions in the literature about whether a chronology protection conjecture is plausible in a homogeneous spacetime-symmetric Gödel universe (this is a reasonably popular question with people studying alternatives to General Relativity such as f(R) and scalar-tensor theories, where there is an adaptation of the null energy condition (NEC); Gödel-solution CTCs are a consequence of the NEC). It also isn't just a little here and a little there. EVERY mass will have a family of observers who see a future image of that mass along with a more usual past image of that mass. For example, an image of a galaxy a couple million parsecs away would be seen by an observer as about the age of observer's host galaxy, while on the same day the same observer sees an image of ~Mpc galaxy aged billions and billions of years older (full of stars much older on average than the ones in the observer's host galaxy). So observer can send a signal to a counterpart in the ~Mpc-distant galaxy giving counterpart billions and billions of years of advanced warning about the details of their galaxy's future configuration.