Immediately after the collision the line joining the 2 points of contact A, B is initially horizontal. Because end A is moving upwards faster than end B the line AB rotates. The angular velocity $\omega$ of the rod is the rate at which line AB rotates. You can find this from $\omega=(v_A-v_B)/l$ assuming the distance AB is $l$.
The angular velocity of a rigid body is the same whichever 2 points are chosen as reference. Pick any two points and measure how fast the line joining them rotates.
Any rigid body motion can be resolved into the translation of a reference point and a rotation about that point. Often we use the CM as the reference point, for convenience. In this case we know the CM follows a parabolic trajectory while gravity is the only external force acting on the rod. But you can use any other point as reference (eg A or B), even one outside of the body.