Algorithm to detect if word belongs to pushdown automaton
As adrianN notes in their comment, your task is certainly algorithmically possible. There are algorithms that take a PDA $A$ and outputs a context-free grammar $G$ such that $L(A) = L(G)$, and others (such as CYK) that take a context-free grammar $G$ and a word $x$ and determine whether $x \in L(G)$.

There might be shortcuts, but at least this shows that your problem is decidable. In fact, since both algorithms run in polynomial time, the problem belongs to $\mathsf{P}$.

Judging from the literature, it seems that this chain of reductions is the standard way of solving the membership problem for pushdown automata. See for example these notes from Hendrik Jan Hoogeboom and Joost Engelfriet.