It seems like time for a mild frame challenge. This habitat is hundreds of kilometres in diameter, and thousands long. Trying to have just one kind of flying machine in such a large environment does not make sense: economics dictates there will be different kinds for different purposes.
If the air pressure on the interior surface is similar to that at sea level on earth, and the spin "gravity" is 1G, then the atmosphere will thin out with altitude in a similar way to Earth's atmosphere, and the axis will be in reasonably hard vacuum. The atmosphere will have to be arranged to rotate with the cylinder. If it did not initially, friction with the interior will spin it up, at the cost of weeks of hurricane-force winds stripping everything off the surface. The atmosphere will naturally form layers, rotating at different speeds, and you will need baffles to avoid permanent rolling storms forming, drawing their energy from the habitat's spin rate, and thus gradually slowing it.
Fixed-wing aircraft will work fine for long-distance travel between different points on the interior surface, as long as they don't try to fly higher than, say, 25Km, where the air starts to become rather thin for them. Aerodynamic lift does not require "real" gravity in any way, it's strictly an interaction between an airfoil and a fluid, producing a force. Helicopters will work fine for shorter distances, jobs that require hovering, and so on.
Current aircraft re-orientate themselves to the curvature of the earth as they fly over it with no problem, and the curvature of the habitat is only about one order of magnitude steeper. Navigation in the more rapidly rotating reference frame will be a bit different from on a planet, but will not be a problem, given simple navigational computers, and an equivalent of GPS. Any society that can reach such a habitat, let along build one, will have no difficulty with the relevant technologies.
Supersonic aircraft might create a noise problem. On Earth, that problem is fairly localised near their flight path, but the concave environment might see shockwaves travelling several times around the cylinder. You need someone who knows more acoustics than me to answer that question.
The more difficult part is reaching the axis region, and loading down ordinary means of transport with the life support and engines needed for that is pointless. Since the axis is effectively in space, you need a spacecraft to survive there. The sensible way to build one is to have "space elevators" at the endcaps, and perhaps at a few points along the length of the cylinder. The latter could run along the cross-beams that will be required to hold the cylinder together. Elevators can take a spacecraft up to the low-gravity, no-atmosphere region, and they can use comparatively low-powered rockets to move around there. Careful navigation will be required, since moving too far from the axis means they'll be accelerated by the upper atmosphere, and start to fall. An emergency parachute system will always be necessary.
Routine travel in the axis region could sensibly use an electric train system with pressurised carriages, and rails arranged so that the trains can't come off them. However, you still need spacecraft for doing repairs to the train tracks, reaching places that don't often need maintenance, and so on.
A good portrayal of the interior of such a habitat, although even larger, can be found in the Schlock Mercenary story "Broken Wind". That habitat had gone rather short of maintenance for several million years, and the spacecraft in the story use super-science gravity generators, but the weather is correct. Since it was constructed by creatures that fly via gas-bags, it lacks a train system, and other devices that surface-dwellers like humans would want.
