I am looking for an example of a closed orientable 4-manifold $M$ with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1(M;\Bbb Z_2)\times H^1(M;\Bbb Z_2)\to H^2(M;\Bbb Z_2)$.
A non-orientable example is $\Bbb RP^4$. An orientable example of dimension 3 is $\Bbb RP^3$.
Note that the squaring operation $H^1(-;\mathbb{Z}/2)\to H^2(-;\mathbb{Z}/2)$ coincides with the Bockstein. It follows your cohomological condition is equivalent to $H_1(M;\mathbb{Z}_{(2)})=\mathbb{Z}/2$. One example of such a manifold is the Enriques surface, which has fundamental group $\mathbb{Z}/2$ and universal cover the K3 surface. More generally, you can construct closed oriented 4-manifolds with any desired finitely presented fundamental group; see the answers to this question, for instance.