This question on space stack begins to tackle how a bird could fly inside an O'Neill cylinder, https://space.stackexchange.com/questions/27665/can-birds-fly-inside-an-oneill-cylinder.
Here is the video mentioned in the question of birds attempting to fly in zero G https://www.youtube.com/watch?v=w4sZ3qe6PiI&feature=youtu.be
The video shows that they can fly in zero gravity, although they seem quite disorientated.
As the above answer points out the birds should fly as normal from take off from the ground where artificial gravity from the spin of the cylinder will take affect like gravity on earth, flying with or against the spin will have interesting effects causing the bird to either dive or hover, but it is the bird flying directly through the zero gravity central axis or along the axis that I am interested in for this question.
How would their flight change as they enter the central zero gravity axis area? and assuming they had lived inside the habitat for some time and had gotten used to the environment. Would they have trouble in this area and avoid it, would they dive through it with their momentum from earlier, could they even take advantage of the area somehow?