This is a good question, that I have no idea how to answer, specifically I have now idea what the answer actually is. I do know that Niven's Ringworld design calls for walls to hold the atmosphere from spilling out into space.
Here's my best guess at an answer:
fact, the "gravity" experienced by any object subject to rotation depends on the speed with which it is travelling
fact, the lower atmosphere will spin with the ring due to friction with it's surface
fact, the energy picked up through friction isn't the going to propagate very far vertically through the air column, on Earth we only see frictional effects in the bottom 300 or so metres of the atmoshpere
Supposition, the lower atmosphere, and everything in it, will experience substantially the same pseudo-gravity as the surface itself but because this gravity is governed by velocity it will dissipate relatively rapidly with altitude. There will also be an atmospheric density below which the gravitational "drop-off" effect accelerates noticeably.
In short something a hundred metres up will probably experience as much if not slightly more gravity than we might expect from a planet but a kilometer or more up it would slow down enough, stellar relative, that it stops experiencing gravitational forces faster than it would on a planet. As such the atmosphere will thin out more rapidly at altitude on a rotating megastructure than on a planet.
I advise anyone looking at building mega-structures to read Larry Niven's essay, Bigger Than Worlds in full as a primer, it has a lot of useful notes.