Some notes on forces affecting the weather.
Tidal forces from the star would affect the atmosphere. There would be very fast and stable wind in the stratosphere. Without bulkheads the dominant wind direction would be opposite to the rotation.
If the bulkheads were high enough they would stop these winds from forming. More probably the bulkheads would be high enough to limit weather but not high enough to stop stratospheric winds. Assuming a composition similar to Earth, this would mean 10km (or more for added safety) high bulkheads. In situation like this I'd assume the top half of the troposphere would be dragged by the stratosphere opposite of the rotation which would have to be balanced by the surface winds in direction of the rotation. The wind speed would depend on the number and height of the bulkheads. So you'd presumably have a near constant wind of the desired speed and stable direction.
One bulkhead would have cool dry air being sucked down. I'd presume the designers would place a sea here to moisturise and warm the air. Since the sea would have the same amount of sunlight as every other part this would work well. Combination of cool dry air above warm moist air would probably create lots of localised thunder storms and even tornadoes, which is another reason to have a sea here instead of habitable land. I don't see large storm fronts or hurricanes being formed, so ships and boats should be able to avoid storms.
The other bulkhead would have warm moist air driven up. This should result in nearly constant rain in the bulkhead mountains. I doubt anyone would want to live in this area, but it would be a good place to build hydro-power.
The area in between would have localized thunderstorms at night, rain on the windward sides of mountains and arid areas behind them. These rain shadows might be stable enough to form deserts, I'd assume arid grasslands exist. Without bulkheads this would be what all of the orbital would be like. Weather would be dominated by the effects of the mountains on the stable wind.
Since rivers and winds would run in opposite directions and both the river flows and the winds would be near constant, sailing ships should be quite practical form of low tech travel. And there would likely be large navigable rivers running from the rains of one bulkhead to the sea at the other bulkhead. In fact, since the winds run at different directions at different altitudes aerial sails or glider aircraft would be much more practical than on Earth as long as you can reach the boundary between the winds. This might actually be important for a low-tech "inheritor" civilization as I doubt orbitals would have huge hydrocarbon reserves. Without bulkheads river directions would be random and wind direction would be same regardless of altitude.
In addition to the tidal and bulkhead effects, there would be effects from the day-night cycle. These should be similar to the effects here on Earth? Although the higher speed the line between day and night moves might make effects negligible outsides coastal areas and areas with large lakes. Basically, air above water warms and cools slower, so there would be winds toward land at the morning and toward water in the evening.
All guesswork obviously...