  We prove that a geometrically integral smooth 3-fold $X$ with nef
anti-canonical class and negative Kodaira dimension over a finite field
$\mathbb{F}_q$ of characteristic $p>5$ and cardinality $q=p^e > 19$ has a
rational point. Additionally, under the same assumptions on $p$ and $q$, we
show that a 3-fold $X$ with trivial canonical class and non-zero first Betti
number $b_1(X) \neq 0$ has a rational point. Our techniques rely on the Minimal
Model Program to establish several structure results for generalized log
Calabi--Yau 3-fold pairs over perfect fields.
