Let $X,Y$ be Noetherian schemes and $f: X \to Y$ is an affine morphism ( $f^{-1}(U)$ is affine for every affine open $U \subseteq Y$ ).
Is it true that $H^1 ( X, \mathcal O_X^{\times}) \cong H^1 ( Y, f_* (\mathcal O_X^{\times})) $ ? If this is not true in general, what if we also assume $f$ is proper (i.e. $f$ is finite) morphism ?
The answer is no for general affine morphisms: take any affine scheme over a field with nontrivial Picard group for $X$ and the spectrum of said field for $Y$. For instance, let $X$ be an affine open in an elliptic curve over some algebraically closed field of characteristic zero.
As written in the question body, the answer is yes for the finite case as discussed here, but you should note that there's a bit of a conflict between the title of your question and the body. The title asks about $Pic(X)$ and $Pic(Y)$, but the body is slightly different, asking about $H^1(X,\mathcal{O}_X^\times)$ and $H^1(X,f_*\mathcal{O}_X^\times)$.