The inclusion operator $\iota \colon c_0\to \ell_\infty$ is strictly cosingular but $\iota^*$ is not strictly singular. Inded, $\iota$ is not weakly compact, so neither is $\iota^*$ (Gantmacher's theorem). As $c_0^*=\ell_1$, by Pełczyński's theorem, $\iota^*$ fixes a copy of $\ell_1$.

The inclusion operator $\iota \colon c_0\to \ell_\infty$ is strictly cosingular but $\iota^*$ is not strictly singular. Indeed, $\iota$ is not weakly compact, so neither is $\iota^*$ (Gantmacher's theorem). As $c_0^*=\ell_1$, by Pełczyński's theorem, $\iota^*$ fixes a copy of $\ell_1$.