Almost 7 years late! Here is my try. Hallo Thorsten!
I call our maps $f \colon \operatorname{Proj} B \to \operatorname{Proj} A$ and $\varphi \colon A \to B$. Surjectivity implies that we actually have a well-defined map $\operatorname{Proj} B \to \operatorname{Proj} A$ and a morphism of schemes in this way.
Being a closed immersion is affine-local on the target. Therefore we can consider some cover of open affines $\bigcup_{j \in J} V_j = \operatorname{Proj} A$ and then check that for each $j \in J$ we have a closed immersion $f \mid_{f^{-1}(V_j)} \colon f^{-1}(V_j) \hookrightarrow V_j$. This is described in Vakil's notes as an exercise.
We have that the collection over all homogeneous $g \in A$ of $D(g) = \{\,p \in \operatorname{Proj} A \mid g \notin p \,\}$ cover $\operatorname{Proj} A$. As $\varphi$ is surjective, we have $f^{-1} (D(g)) = D(\varphi(g))$.
We now have
\begin{align*}
f \mid_{D(\varphi(g))} \colon D(\varphi(g)) & \hookrightarrow D(g) \\
p & \mapsto \varphi^{-1} (p) \, .
\end{align*}
These sets are all open affines! For any graded ring $R$, we have for any homogeneous $h \in R$ the identification $D(h) = \operatorname{Spec}(R_h)_0 = \operatorname{Spec}\{\, \frac{x}{h^n} \mid n \in \mathbb N, \, \deg x = \deg h \cdot n \,\}$. (Sometimes, $(R_h)_0$ is confusingly written as $R_{(h)}$.) Our map can then be seen as
\begin{align*}
f \mid_{\operatorname{Spec} (B_{\varphi(g)})_0} \colon \operatorname{Spec} (B_{\varphi(g)})_0 & \hookrightarrow \operatorname{Spec} (A_g)_0 \\
p & \mapsto \varphi^{-1} (p) \; ,
\end{align*}
which corresponds to the surjective ring homomorphism
\begin{align*}
\varphi (D(g)) \colon (A_g)_0 & \to (B_{\varphi(g)})_0 \\
\frac{x}{g^n} & \mapsto \frac{\varphi(x)}{\varphi(g)^n} \; ,
\end{align*}
which means that $f \mid_{f^{-1}(D(g))} \colon f^{-1}(D(g)) \hookrightarrow D(g)$ is a closed immersion, concluding that $f$ is a closed immersion.