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Maybe the following works?

Let's consider the ringed space $(X, \pi_* O_{\widetilde X})$ and also the ringed space $(\widetilde X, O_{\widetilde X})$. Certainly $\pi_{\*}$ gives us an isomorphism $$ \text{Pic}(\widetilde X, O_{\widetilde X})) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) $$ which sends an $O_X$-line bundle $L$ to the $\pi_* O_X$-module $\pi_* L$, right? (I'm using the fact that $X$ is projective here to show that there exist open trivializing sets containing various finite collections of points). Also observe that $(\pi_* O_{\widetilde X})^{\*} = \pi_* (O_{\widetilde X}^{\*})$. Thus by Hartshorne Chapter III, Exercise 4.5 (which works on arbitrary ringed spaces), we see that $$ \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_* O_{\widetilde X})^*). $$ Combining with the isomorphisms already written, we obtain: $$ \text{Pic}(\widetilde X, O_{\widetilde X}) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_{\*} O_{\widetilde X})^*) \cong H^1(X, \pi_{\*} (O_{\widetilde X}^{\*})) $$

Maybe the following works?

Let's consider the ringed space $(X, \pi_* O_{\widetilde X})$ and also the ringed space $(\widetilde X, O_{\widetilde X})$. Certainly $\pi_{*}$ gives us an isomorphism $$ \text{Pic}(\widetilde X, O_{\widetilde X})) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) $$ which sends an $O_X$-line bundle $L$ to the $\pi_* O_X$-module $\pi_* L$, right? (I'm using the fact that $X$ is projective here to show that there exist open trivializing sets containing various finite collections of points). Also observe that $(\pi_* O_{\widetilde X})^{*} = \pi_* (O_{\widetilde X}^{*})$. Thus by Hartshorne Chapter III, Exercise 4.5 (which works on arbitrary ringed spaces), we see that $$ \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_* O_{\widetilde X})^*). $$ Combining with the isomorphisms already written, we obtain: $$ \text{Pic}(\widetilde X, O_{\widetilde X}) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_{*} O_{\widetilde X})^*) \cong H^1(X, \pi_{*} (O_{\widetilde X}^{*})) $$

Maybe the following works?

Let's consider the ringed space $(X, \pi_* O_{\widetilde X})$ and also the ringed space $(\widetilde X, O_{\widetilde X})$. Certainly $\pi_{\*}$ gives us an isomorphism $$ \text{Pic}(\widetilde X, O_{\widetilde X})) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) $$ which sends an $O_X$-line bundle $L$ to the $\pi_* O_X$-module $\pi_* L$, right? Also observe that $(\pi_* O_{\widetilde X})^{\*} = \pi_* (O_{\widetilde X}^{\*})$. Thus by Hartshorne Chapter III, Exercise 4.5 (which works on arbitrary ringed spaces), we see that $$ \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_* O_{\widetilde X})^*). $$ Combining with the isomorphisms already written, we obtain: $$ \text{Pic}(\widetilde X, O_{\widetilde X}) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_{\*} O_{\widetilde X})^*) \cong H^1(X, \pi_{\*} (O_{\widetilde X}^{\*})) $$

Maybe the following works?

Let's consider the ringed space $(X, \pi_* O_{\widetilde X})$ and also the ringed space $(\widetilde X, O_{\widetilde X})$. Certainly $\pi_{\*}$ gives us an isomorphism $$ \text{Pic}(\widetilde X, O_{\widetilde X})) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) $$ which sends an $O_X$-line bundle $L$ to the $\pi_* O_X$-module $\pi_* L$, right? (I'm using the fact that $X$ is projective here to show that there exist open trivializing sets containing various finite collections of points). Also observe that $(\pi_* O_{\widetilde X})^{\*} = \pi_* (O_{\widetilde X}^{\*})$. Thus by Hartshorne Chapter III, Exercise 4.5 (which works on arbitrary ringed spaces), we see that $$ \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_* O_{\widetilde X})^*). $$ Combining with the isomorphisms already written, we obtain: $$ \text{Pic}(\widetilde X, O_{\widetilde X}) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_{\*} O_{\widetilde X})^*) \cong H^1(X, \pi_{\*} (O_{\widetilde X}^{\*})) $$

Maybe the following works?

Let's consider the ringed space $(X, \pi_* O_{\widetilde X})$ and also the ringed space $(\widetilde X, O_{\widetilde X})$. Certainly $$ \text{Pic}(X, \pi_* O_{\widetilde X}) \cong \text{Pic}(\widetilde X, O_{\widetilde X})). $$ Also observe that $(\pi_* O_{\widetilde X})^{\*} = \pi_* (O_{\widetilde X}^{\*})$. Thus by Hartshorne Chapter III, Exercise 4.5 (which works on arbitrary ringed spaces), we see that $$ \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_* O_{\widetilde X})^*). $$ Combining with the isomorphisms already written completes the proof.

Maybe the following works?

Let's consider the ringed space $(X, \pi_* O_{\widetilde X})$ and also the ringed space $(\widetilde X, O_{\widetilde X})$. Certainly $\pi_{\*}$ gives us an isomorphism $$ \text{Pic}(\widetilde X, O_{\widetilde X})) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) $$ which sends an $O_X$-line bundle $L$ to the $\pi_* O_X$-module $\pi_* L$, right? Also observe that $(\pi_* O_{\widetilde X})^{\*} = \pi_* (O_{\widetilde X}^{\*})$. Thus by Hartshorne Chapter III, Exercise 4.5 (which works on arbitrary ringed spaces), we see that $$ \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_* O_{\widetilde X})^*). $$ Combining with the isomorphisms already written, we obtain: $$ \text{Pic}(\widetilde X, O_{\widetilde X}) \cong \text{Pic}(X, \pi_* O_{\widetilde X}) \cong H^1(X, (\pi_{\*} O_{\widetilde X})^*) \cong H^1(X, \pi_{\*} (O_{\widetilde X}^{\*})) $$