EDIT:
I managed to solve it by defining $v(x):=\frac{C_1}{2}(x_1-\frac{1}{2})^2+C_2$ which is subharmonic and satisfies $v\geq u$, then generalizing $v$ by setting the constants to be the maxima of $f,g$ on their respective domains.
Let $Q:=(0,1)^n, f \in C^{0}(\overline{Q}),g\in C^{0}(\overline{\partial Q}).$ Furthermore $u \in C^{2}(Q)\cap C^{0}(\overline{Q})$ is a solution to $$\begin{cases} -\Delta u = f & x\in Q\\u=g & x\in \partial Q \end{cases}$$ Prove: $\| u \|_{L^{\infty}(Q)} \leq \frac{1}{8}\|f \|_{L^{\infty}(Q)}+\|g \|_{L^{\infty}(\partial Q)}.$
So I first wanted to figure out the more simple case, where $f=C_1,g=C_2>0$ constant. If I could construct a non-constant function $v\geq u$ (that is subharmonic and inherits the properties from $u$) I could apply the strong maximum principle and get an upper bound for $\|v \|_{L^{\infty}(Q)}$ involving just a constant (the maximum), which if still bounded by $\frac{1}{8}C_1+C_2$ would yield the desired result in the simplified case.
Maybe I can then apply my findings in the more general case. Unfortunately I am lost as to how to find such a function $v$. I'd be grateful for any tips or sketches of this proof.