  In this note, we compute the virtual first Betti numbers of 4-manifolds
fibering over $S^1$ with prime fiber. As an application, we show that if such a
manifold is symplectic with nonpositive Kodaira dimension, then the fiber
itself is a sphere or torus bundle over $S^1$. In a different direction, we
prove that if the 3-dimensional fiber of such a 4-manifold is virtually fibered
then the 4-manifold is virtually symplectic unless its virtual first Betti
number is 1.
