We say that an element $u$ in a $C^{*}$-algebra is unitary if $u^{*} =u^{-1}$. Furthermore we assume $u$ is normal as well And because of that $$\sigma(u^{-1}) = \{ \lambda^{-1} : \lambda \in \sigma(u)\} = \{ \overline{\lambda} : \lambda \in \sigma(u) \}. $$
To me that inequality does not seem obvious, where does it come from?
Those equalities have nothing to do with normality.
In any case, the equality $u^*=u^{-1}$ implies that $u$ is normal, as any element commutes with its inverse by definition.
Note first that since $u$ is invertible we may assume $\lambda\ne0$.
The equality $$ (u-\lambda) ^{-1}=-u^{-1}\lambda^{-1}\,(u^{-1}-\lambda^{-1})$$ shows that $$\sigma(u^{-1}) =\{\lambda^{-1}:\ \lambda\in\sigma (u) \}. $$ Similarly, the equality $$(u^*-\overline\lambda) ^{-1}=[(u-\lambda)^{-1}]^*$$ implies $$\sigma(u^{*}) =\{\overline\lambda:\ \lambda\in\sigma (u) \}. $$