I thought of a different example which still uses some machinery but is of a rather different nature. Let $S$ be the Enriquez surface. Then $X = S \times \mathbb{CP}^1$ is uniruled, hence has Kodaira dimension $-\infty$. But $b_{1}(X) = 0$, hence $h^{0,1}(X)=0$.  Then we can can conclude that $X$ is not rational or Fano since for example $\pi_{1}(X) = \mathbb{Z}_{2}$.