If we imagine a river that circles the habitat around the axis of rotation I think such a river will have an apparent flow from the perspective of an observer standing on the banks. I will explain my reasoning below.
The first thing we need to address is L.Dutch’s criticism that there must be an energy source to generate flow. As Molot pointed out in the comments that energy can come from the rotational spin of the habitat. In fact, any nonrigid body will lose speed due to friction and turbulence in the gases and liquids inside it. What this means is that the atmosphere and hydrosphere inside the habitat will slow in relation to the rigid surface of the cylinder. Friction between the rigid and nonrigid will speed up the water and air and slow down the ring. This is a constant process that will gradually slow the ring. This means that at equilibrium between the two opposing frictional forces we can expect that on average the nonrigid components of the system will have a slightly longer rotational period than the rigid components. That is to say, the air and water will tend to have an anti-spinward velocity from an observer standing on the inner surface of the ring.
This effect is minor but I think it will be exacerbated by two additional forces. The first is a part of the Coriolis effect. If we look at HDE226868’s answer we see that the Coriolis effect on our river is a vertical one, rather than a horizontal one, but because our river has a vertical this will still effect the flow of the river.
On Earth, a train going around the equator to the East (spinwards) is lighter than a train going West (anti-spinwards). This is due to the vertical component of the Coriolis effect called the Eötvös effect. Essentially, the centrifugal force of the Earth’s spin acts against the pull of gravity and tries to fling us into space. Spinning faster increases this force and makes us even lighter while spinning more slowly reduces this force and makes us heavier.
On Earth, this effect is slight and only important for rocket launches and long-range artillery bombardment, but on our relatively small spinning habitat, the magnitude would be much larger. Now how does this apply to our river? Because our spinning cylinder is “inside-out” compared to the Earth the forces are reversed. Water moving in a spinward direction faster than the water around it will be effectively heavier and water moving in an anti-spinward fashion will be lighter. This means that spinward currents will sink and anti-spinward currents will rise. This will result in the surface of the river having a larger anti-spinward velocity than the bottom of the river. In this way, the Eötvös effect will exacerbate the perceived flow of the river from an observer on the surface.
The second effect is that of wind. I anticipate that the wind at the surface of the habitat will be primarily anti-spinward and that this, as a result, will act to pull the river further anti-spinward. My reasoning is as follows. All of the aforementioned effects are acting on the air of the habitat just as they were the water. This means the air will also have a net anti-spinward velocity relative to the ring, with higher altitudes having larger anti-spinward velocities. Additionally, the heat cycle will play a role here. Hot air on the surface of the ring heated by the artificial sun will rise due to decreased density just as it does on Earth. However, here the Coriolis effect will deflect the rising air spinwards. In turn, the cool air from above that sinks to take the warm air’s place will be moving anti-spinward. In this way, convection currents on the rotating habitat will create strong anti-spinward winds on the surface of the ring. The surface of any water will, therefore, be pushed in an anti-spinward direction by the wind.
These forces, the various frictions and Coriolis effects, will act together to cause the surface of a circular river to flow anti-spinward in an endless cycle powered by the rotational kinetic energy of the system which will be gradually lost to heat.