Estimate solution of Poisson equation on unit ball

Let $w(x)=a(|x|^2-1)$, where $a$ is a constant whose value will be adjusted shortly. By construction, we have $$ w|_{\partial U} = 0. $$ On the other hand, we compute $$ \Delta w = 6a, $$ and so by choosing $$ a=\frac16\inf_Ug, $$ we ensure $$ \Delta u\geq\Delta w\qquad\textrm{in}\,\, U. $$ Thus the comparison principle gives $$ u\leq w\qquad\textrm{in}\,\, U, $$ that is, $$ u(x)\leq\frac{|x|^2-1}6\inf_Ug. $$

Yes you can. In fact, there is a theorem :

Theorem (a priori bound) : Let $u\in C^2(\Omega)\cap C^0(\overline{\Omega})$ be a solution of the Dirichlet problem $Lu=f$ in $\Omega$, $u=g$ on $\partial\Omega$, with $\Omega$ bounded, and $Lu=-\sum_{i,j=1}^na_{ij}(x)\partial_{x_i,x_j}u(x)+\sum_{i=1}^nb_i(x)\partial_{x_i}u(x)+c(x)u(x)$ is uniformly elliptic with $c\geq 0$. Then $$||u||_{C^0(\overline{\Omega})}\leq||g||_{C^0(\partial\Omega)}+C\sup_{x\in\Omega}\frac{|f(x)|}{\lambda(x)}$$ with $C=e^{2(B+1)}-1$, and $B:=\sup_\Omega \frac{|b(x)|}{\lambda(x)}$.

Where uniform ellipticity of the operator is defined as :

Definition : $L$ is said to be elliptic in $\Omega$ if there exists positive functions $\lambda,\Lambda :\Omega\rightarrow \mathbb{R}_+$ such that $$0<\lambda(x)|\xi|^2\leq \xi^TA(x)\xi=\sum_{i,j}a_{i,j}(x)\xi_i\xi_j\leq\Lambda(x)|\xi^2|,$$ for all $x\in\Omega$, $\xi=(\xi_1,\cdots,\xi_n)\in\mathbb{R}^n$. $L$ is uniformly elliptic if there exists $\lambda_0,M\in\mathbb{R}$ such that $$\lambda(x)\geq \lambda_0 >0\text{ and }\frac{\Lambda(x)}{\lambda(x)}\leq M\leq \infty,$$ for all $x\in\Omega$.

So In this case, (by the other theorem we know the solution exists) $g=0$, $\Omega=S^3$, and $L=\Delta$, which is uniformly elliptic with $\lambda(x)=1$. Therefore, we have a priori bound : $$||u||_{C^0(\overline{\Omega})}\leq (e^2-1)||g||_{C^0(\Omega)}.$$ But this bound covers all $x\in\Omega$, so almost surely not a optimal bound for $x_0$. Similar to @timur did, you can have a super solution that is explicit, and try to get a better bound.