The full problem of an O'Neill cylinder is pretty tough to solve.
1. A simple approach
Analytically, one would assume to be in hydrostatic equilibrium, without fluid motion on the surface of the O'Neill cylinder. Then one can use the equations of planetary atmospheres in cylindrical coordinates to derive an approximation to the pressure profile.
The radial velocity equation on the surface $z=z_0=r_0$ with boundary condition $v_{\rm cylinder} = \Omega_0 r_0$ becomes
$$\Omega^2_0 r = -\frac{1}{\rho}\partial_rP$$
Which in the simple, isothermal case with Temperature $T$ and corresponding sound speed $c^2_s = k_B T /\mu$, with boundary condition $P(r) = P_0$ has the solution
$$P(r) = P_0 \exp(-\frac{\Omega_0^2}{c^2_s}\frac{r_0^2-r^2}{2})$$
where we keep in mind that the radial coordinate and the vertical one are connected via $r=r_0-z$.
Thus, one gets a gaussian pressure profile with a modified scale-height $H=c_s/\Omega_0$ in this, the simplest case.
It is instructive rewrite this scale-height as ratio of velocities,
$$H=\frac{c_s}{\Omega_0} = \frac{c_s}{v_{cylinder}} r_0 $$
because then we can relate scale-height to cylinder radius. I take $c_s = 350m/s$ like in an Earth-like atmosphere and $v_{cyl}$ becomes with your data $v_{cyl}=83m/s$, so we see that the scale height is very large compared to the cylinder radius, thus there wouldn't be any strong density stratification in the cylinder.
Then, dynamical effects can have a huge impact over the dynamics in the cylinder. Through friction, the rotating surface will inject lateral momentum into the atmosphere, forcing it to co-rotate. At the center of the cylinder nothing can co-rotate. Thus clearly the situation is unstable, and there will be some adjustment of the pressure profile.
2. The full approach and the rotation profile
One would have to solve the set of vector equations
$$(\vec v \cdot \vec \nabla)\vec v + 2 \vec \Omega \times \vec v + \Omega^2 \vec r_{\perp} = -\frac{\vec \nabla P}{\rho} + \frac{\nu}{\rho} \Delta
\vec v$$
where only gravity is missing now.
The question you've had whether the air would rotate en masse as a solid body, or whether there would be shear, depends strongly on how effective turbulent viscosity would is in transporting momentum from the wall into the atmosphere. I don't know the answer to that, I think one would need to simulate this.
My intuition tells me that upwards momentum transport could be much more effective than on Earth, due to the weak stratification. But even then one needs some type of instability for the turbulent vortices to pump momentum into the systematic rotational velocity of the gas. So it might as well be that the gas will simply destabilize and breakup into large vortices in the radial-rotational direction.
3. In the book
Rama starts out much colder, so what I'm speculating is that the author has done the same little solution as we have above and said "oh I'll do a transition from strong to weak stratification by letting $c_s<v_{cylinder}$ initially and then $c_s>v_{cylinder}$ which should then create a storm".
Yeah, so far my answer.
