Would there be any gravity inside a hollow planet made of a spherical shell of basalt crust?

If you neglect the gravity of the nitrogen gas, there is no net gravitational force anywhere inside. This is a result of the famous “Shell Theorem” for inverse-square forces. At any point, you are attracted by all the atoms in the shell, but the vector sum of all these forces — in different directions, and having different magnitudes because they are caused by atoms at various distances from you — turns out to be zero. This assumes a perfectly spherical shell of uniform mass per unit area.

If you want to take the nitrogen gas into account, at radius $$r$$ from the center you are attracted by the mass of all the nitrogen in a sphere of radius $$r$$ around the center, as if this mass were concentrated in a point at the center. The nitrogen at larger radii doesn’t attract you for the same reason that the shell doesn’t.

To add to @GSmith comment, if you take into account the nitrogen inside, you would only feel the gravitational pull of the nitrogen between you and the center of the planet (as explained). It is easy to see that anyone who falls inside will end up right at the centre of the planet:

1. You begin to fall towards the centre of the planet due to the pull of the nitrogen inside.
2. Since you can go through nitrogen (its not solid but gas!) you will go past the centre out towards the other end of the planet.
3. This oscillatory motion would continue for ever would it not be for the air resistance of the nitrogen which keeps substracting speed/energy from you. This means that with every swing about the centre you reach a lower altitude each time (just like in a kid's swing) until eventually you end up in centre. Think of this motion like a real pendulum which ends up stopping at the middle but in a 3D kind of way. What a way to die!