It is decidable whether a pushdown automaton will accept a word?
Pushdown automata are nondeterministic, and there is no a priori bound on the length of computation (imagine $\epsilon$-transitions that add and remove the same element from the stack). Therefore it's not clear how to perform your simulation.

Fortunately, we know how to convert a pushdown automaton to an equivalent context-free grammar, and the latter have parsing algorithms such as CYK. Using these algorithms, we can dedice whether a given pushdown automaton accepts a word by first converting the automaton into a context-free grammar.