This article talks about a new theory where the big bang happened in one direction for us and the other direction where time is going backwards (mathematically). Would that revive the idea of a cyclical universe again? I had this little vision in my head when I read this. A universe chain with a big bang on essentially on either side of the start/end of time and in the next big bang our universe is the one going backward in time and the new one is going forward.
Perhaps it does. But I would also stipulate that the two-sided inversion is also overlaid, meaning that we cannot disconnect from the other side, meaning that we need a mirror side of reality to make sense of existence, and meaning that there are likely lesser mirrors (e.g., an actual mirror, the condition called situs inversus, or CP symmetry that appears disconnected from CPT symmetry that is more fundamental)that are somewhat removed from the must fundamental mirror.
Your link refers, first, to a paper by Carroll & Chen, whose complete version can be seen at https://arxiv.org/abs/hep-th/0410270 . Vilenkin criticized it in a paper at https://arxiv.org/abs/1305.3836, providing reasons ("under very mild assumptions", which he details) to explain why the (local) universe it attempted to describe would be literally "surrounded by singularities", which would prevent it from sharing its Big Bang / Big Crunch with another universe.
The link did not address these difficulties, but, instead, provided another (completely different) possibility based upon an extremely local experiment by the mathematician Barbour, which, as stated, relied upon Newtonian gravity. Newtonian gravity is entirely adequate for plotting orbits within the solar system, but the fact that it has never sufficed to explain the precession of Mercury's orbit (which is a very slow rotation of the orbit's axis around a point very far outside our system) shows that it will not suffice for any complete cosmology: Actually, as described by Guth with simple algebra on p.296-297 in the 1997 edition of his book titled "The Inflationary Universe", "an infinite distribution of matter under the influence of Newtonian gravity would unquestionably collapse", and, in fact, would collapse as soon as its expansion might've otherwise begun. I imagine Barbour (perhaps the world's foremost mathematician) may have accepted Newtonian gravity for the experiment only because of the current uncertainty about the interaction between gravity and subatomic particles.
It IS possible to hypothesize local universes that do not contain any matter at all, and it is possible that such universes comprise most of reality, but the illustration provided by the link showed the usual assortment of stars within the ones it was hypothesizing.
Concerning cosmologies describing local universes, containing matter during some part of an existence that might either continue indefinitely or recur more-or-less periodically, I've analyzed two (one of them similar to Carroll & Chen's, but a little older and simpler) at Is the Universe Past-Eternal? .
What I believe they are getting at is a theory that, before the Big Bang, there was a universe that was eternally contracting, and then it collapsed into a singularity and then exploded again to form our universe which now will be eternally expanding.
Hence, the "mirror" universe lies, not in some "parallel dimension" to use a kind of mathematically awkward pop-cult term, but in our past, so if you wanted to "go there", you'd have to go back in time. You'd need a time machine to get into it, since it's more than 13.8 billion years (437 Ps) ago. Unless and until someone invents a flux capacitor and gives it 1.21 jiggawatts (JW) [not to be confused with gigawatts (GW), by the way], or perhaps more even, given this is so far back, that ain't happening :)
Now, what was it like in this "mirror" universe? Well, it would likely have been just like ours but, if you view "ours" and its time as proceeding the same space-time axis, what you would see there would be that everything was running backwards: entropy would be ever-decreasing, not increasing. People would walk backwards, stars would be sucking up light, black holes would expand into stars (meaning they would actually be white holes, formed by collapsing Hawking radiation tuned with impeccable rightness to form a white hole for real and not just in sci-fi), rot gas would turn into corpses and dead people would come back to life and so forth as it all winds back to the Gnab Gib which then becomes the Big Bang - but likely not the same people, aliens and other such things as in our Universe and, for them, they would interpret this as a Big Bang and time going normally because their entropy arrow, which determines the direction of their conscious peception, goes the other way and so to them, "we" "already happened" and "our" Big Bang was their Gnab Gib, and someone there is talking goofily about some version of sroticapac xulf whose purpose is to go, from our perspective, forward in (our sense of, backwards in theirs) time to our world, etc. Two universes "joined at the hip", so to speak.
Or to imagine it more simply: Take a movie - any movie. Take a second movie. Reverse it so it plays backwards. Concatenate the first movie to the end of the reversed version of the second movie. Play this pair of movies connected together which will be joined at their beginnings as one single movie. That is what this is like, only on a universe-wide scale.