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 $h^{0,1}(X)=0$. On the other hand $\pi_{1}(X) = \mathbb{Z}_{2}$. Then we can can conclude that $X$ is not rational or Fano by the following 2 facts.
Fano Varieties are simply connected (either using Yaus solution of the Calabi Conjecture or Moris Bend and Break).
Rational varieties are simply connected, since $\pi_1$ is a birational invariant and $\pi_{1}(\mathbb{CP}^3)=\{1\}$.