Travel Counter-Spinward would be faster then Spinward as the rate of rotation on such a small diameter would be noticeable.
Plotting a course directly point to point through the middle of the centre axis would cause an acceleration effect as the pull of gravity reduced only to slow down again as you near the opposite side.
In most sci-fi where this kind of habitation has been used there is usually a light giving source down the central axis of the cylinder to act as a kind of "sun" lighting up the whole inner surface of the cylinder. This would negate the possibility of flying through the central axis.
The main danger is from effectively flying inside of a box with gravity, the ground is on every side of you. So at any one time you will be flying with a ground collision possibility should you maintain level flight and steer in any one direction for too long without initiating a correction spin and a fast climb can soon turn into a fast dive !
In reference to the ground there would be no way to fly a straight course from point A to point B axially, the course would end up being a curve as the terrain underneath you rotates away.
One interesting effect would be to fly straight up to the middle of the central axis, wait a few minutes and then drop straight back down again as the land below you has rotated away.
Instead of thinking of conventional aircraft from a planetary perspective of living on the outside of a ball with gravity and atmosphere think more along the lines of gyro-stabilised rotation.
you could have something like a quad copter with a central compartment and props that always stayed in relative horizontal reference to the "ground" but independently of each other. So whilst the prop assemblies rotate around the outside in a full 360 x 360 x 360 arc ( X,Y,Z or Pitch, Roll and Yaw) the cockpit can do the same in its own 360 x 360 x 360 arc. This would make flying far easer and much more aerobatic to counter the special needs within this kind of environment :o)