Artificial gravity in movie Interstellar

Cooper Station is somewhat like a hollow cylinder. Buildings are attached on the surface.

It has artificial gravity to hold on buildings and people. If, a person needs to fly in this spot

Will the airplane experience two different gravitational field?

The centrifugal force at distance $$r$$ from the station centre is then $$g=\left(\frac{2π}T\right)^2r$$. When something is close to the axis, like your plane, $$r≃0$$ and the gravity is negligible. The station rotates around the plane, which does not feel any gravity.
Assuming the station has a 800 m radius (as computed in this scifi.sx answer) and a Earth gravity $$g≃10\:\mathrm{m}⋅\mathrm{s}^{-2}$$, it makes a turn every $$T=2π\sqrt{\frac{r}{g}}≃2π\sqrt{\frac{800}{10}}\:\mathrm{s}≃18π \:\mathrm{s}\simeq 54 \:\mathrm{s}.$$ With a turn every minute like this, one could feel the Coriolis force quite quickly. If one moves along the station axis, it will lead to an acceleration of $$\frac{2\pi}{T}v$$ for someone moving at speed $$v$$. An athlete running at $$v=9\:\mathrm{m}⋅\mathrm{s}^{-1}$$ will feel a lateral acceleration of $$10\%$$ of the gravity. For anything moving faster than $$90\:\mathrm{m}⋅\mathrm{s}^{-1}= 324\mathrm{m}/\mathrm{h}=202\:\mathrm{mph}$$, like a plane, the Coriolis force will be stronger than gravity, and it will imply strange trajectories.