There's an extensive (and I mean extensive) essay by Gregory Benford on the Centauri Dreams website regarding the physical mechanics of The Bowl in his novels 'Bowl of Heaven' and 'Shipstar'.
In short, like a classic Ring world, the atmosphere is relatively thin (albeit thicker than our own) and is held in place by centrifugal action.
The atmosphere is quite deep, more than 200 km. This soaks up solar
wind and cosmic rays and makes the Bowl toasty through greenhouse
effect. Also, the pressure is higher than Earth normal by about 50%,
depending on location in the Bowl. It is also a reservoir to absorb
the occasional big, unintended hit to the ecology. Compress Earth’s
entire atmosphere down to the density of water and it would only be 30
feet deep. Everything we’re dumping into our air goes into just 30
feet of compressed nitrogen and oxygen, then. The Bowl has much more,
over a hundred yards deep in equivalent water. Too much carbon
dioxide? It gets more diluted.
Since the plan is to view something approximately the size of a mountain range through 400KM of atmosphere, at a distance of
one hundred million miles
(roughly twice the distance from Earth to Mars) you would need a truly vast telescope unless the cities were positively pouring out waste light. The best Earth-based telescopes can distinguish large surface masses on Mars and easily distinguish the icy poles but that's about it. You need to be in space to get the kind of hubble-clear images that we enjoy.
Also, it's worth mentioning that if you're trying to view something directly through the plume (e.g. right on the other side of the bowl) , it would be that much harder, probably impossible due to the light haze.
In short, it's feasible to view megastructures on the other side of The Bowl (with a large enough telescope and enough technical know-how to counter the atmospheric interference) but it's a very inelegant way of doing so and would require the people on the other side to have advanced well beyond the technology needed for what is essentially a very big semaphore tower.