The answer is no. As was described in the thread: Spin structure on mapping torus
a mapping torus has a spin structure if and only if the monodromy of the bundle (over $S^1$) fixes a spin structure. The mapping torus I'm going to describe is a bundle over $S^1$ with fibre $S^1 \times S^2$.
The mapping class group of $S^1 \times S^2$ is of order $8$. If you pass to the subgroup that preserves the fundamental classes of the factors, you get a subgroup of order $2$, the generator can be thought of as the diffeomorphism that twists the $S^2$ factor by $2 \pi$ as one walks around the circle factor.
This automorphism does not preserve any spin structure on $S^1 \times S^2$, it acts as an involution on the spin structures, with no fixed spin structures.