Geodesics in general relativity changing from timelike to spacelike? Is it possible for geodesic in general relativity to be time-like, and at some point cross null-plane and become space-like?
 A: 
Is it possible for geodesic in general relativity to be time-like, and at some point cross null-plane and become space-like?

No. One way of defining a geodesic is that it parallel-transports its own tangent vector. Parallel transport preserves the norm, so it can't change a timelike vector to a spacelike one.
A: 
Tomek asked: Is it possible for geodesic in general relativity to be
  time-like, and at some point cross null-plane and become space-like?

As Ben Crowell already said, that is not possible. After you crossed the horizon of a black hole your velocity relative to the singularity will indeed exceed the speed of light, but you would have to be on a spacelike path in order to keep a constant distance behind the horizon, or on a lightlike path to stay stationary at the horizon, which you can't because as a massive observer you are forever confined to a timelike path. Also you can never overtake a photon, neither outside nor inside of the horizon. Therefore on a Penrose diagram the path of a massive particle must always be inside the light cone and have an angle steeper than 45°, inside and outside the horizon. To cite MTW, exercise 13.5: 

MTW wrote: A geodesic which is timelike at one event is everywhere
  timelike. Similarly, a geodesic initially spacelike is everywhere
  spacelike, and a geodesic initially lightlike is everywhere lightlike.

