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Higher floors would experience lower gravity. Walking with the spin would slightly increase the apparent force of gravity, while walking against the spin would reduce it. Dropped or thrown objects would follow curving paths due to Coriolis forces. Since the floors of the buildings are flat instead of cylinder sections, gravity would also tend to pull things toward their east and west edges, especially on upper floors.
Also, because the cylinder is rotating around its long axis and contains flexible objects, it wants to shift its axis and rotate end-over-end instead, which exerts a crosswise torque on the whole ship that has to be counteracted somehow.
The cylinder is both tapered at the ends, and have a separate system, using water reserves of the ship, that spins in the opposite direction (well, pumps water, to be more technical) to counteract the spin force the cylinder excretes on the ship's non-rotating frame (In addition, the mass of the "flexible objects", the atmosphere, is insignificant in comparison to the rest of the drum). It's a quite complex and self-correcting system, which, combined with the size, explains why there isn't many ships of that class around.
Upper parts of the building are a mix of technical floors and rooms that make use of the reduced but non-zero gravity. As to what you would experience, well, depends on the speed. I once rode on an elevator that descended very fast. It was... an interesting experience. Sort of almost falling.
Thank you everyone for the interesting discussion.
Maybe, a large volume of water (eventually in three phase equilibrium, liquid water, ice, and vapor) could also act as a "thermal flywheel" for environmental control.
There are few science-fiction web comics that I know of, which stick with "hard" (realistic) science. Recently on Freefall by Mark Stanley, the protagonists are visiting a spinning space station. (Hint: check which way the station is rotating before pouring yourself a cup of coffee!)