Well, presumably you know that the volume of a sphere is given by and that of a cylinder by . Here, you are told that R= 4 and that r= 2. To find the length of the cylinder, imagine lines from the center of the sphere to where the cylinder intercepts the sphere, which is a radius of the sphere and so has length R, and a line from the center of the sphere perpendicular to the axis of the cylinder, which is a radius of the cylinder and so has length r. Those two lines from the hypotenuse and one leg of a right triangle with the half the length of the cylinder the other leg. By the Pythagorean theorem, .
Now, at this point, you have to be careful. The "volume inside the sphere and outside the cylinder" is NOT the volume of the sphere minus the volume of the cylinder. That ignores the two "caps" at each end of the cylinder that will also be removed from the sphere but are beyond the length of the cylinder calculated above. Wikipedia gives a formula for finding the volume of such "spherical caps": Spherical cap - Wikipedia, the free encyclopedia