The simple answer to this is "yes, go read the paper". Indeed, the OP is somewhat negligent in their post, since the Celi et al. paper plainly states in the introduction that
We also show that by using additional Raman and radio-frequency transitions one can connect the edges in the extra dimension.
Thus, the possibility of engineering synthetic cylinders was on the cards to begin with.
However, later work does propose much fancier topologies for these sorts of schemes. More specifically, in
Quantum simulation of non-trivial topology. O. Boada et al. N. J. Phys. 17 045007 (2015) (open access)
the ICFO group proposes a number of interesting schemes, including
a Möbius strip,
and a twisted torus,
Of course, whether these schemes can be realized experimentally is another matter. In particular, they require very specific hyperfine couplings to be turned on, while other couplings are not. This is in principle OK but it does require a good bit of fine-tuned control. Additionally, some of these schemes require sites to be individually targeted, which can be very hard, particularly if the links are directly implemented in the microwave domain.
As an example of the difficulties, consider the implementation by Mancini et al.. They used Raman transitions for the hyperfine couplings, as they (i) are generally easier to implement than direct microwave transitions, and (ii) have a much larger spatial phase variation, which is required to implement the large Peierls phases they needed to simulated the synthetic magnetic field. However, Raman transitions consist of two dipole transitions instead of one, which forces $\Delta m=2$ between lattice sites instead of $\Delta m=1$, and therefore cuts in half the available lattice width (i.e. three sites instead of six, with a spin-5/2 system).
An additional point worth mentioning is that often you don't really even want such fancy topologies in the first place. This is the case, for example with the Mancini et al. paper, where they care about chiral edge states. These are cyclotron orbits in a magnetic field, which can only exist in a single direction near a sharp edge:
Thus, for these purposes it is vital to have a sharply defined cutoff, and it is advantageous to have a thin strip (as long as it is wide enough to support the cyclotron orbits), as this maximizes the relative contribution of the edge states.
For other interesting effects, of course, you probably do care about the topology. To see what signatures you might look for, and why you'd care about them, see the Boada et al. paper.