Let's abstract from the random walk formulation, as you first have to specify what do you mean by the random walk on a bounded interval. In any case, it will be an example of an irreducible finite-state Markov Chain $\Phi$. Let $\pi$ be the invariant *probability* distribution of $\Phi$. We know that for any state $j$:
$$
\pi(j) = \frac{1}{\mathsf E_j[\tau_j]} \ ,
$$
see e.g. here, so
$$
\mathsf E_j[\tau_j] = \frac{1}{\pi(j)}
$$
and it does make sense to estimate $\tau_j$ as its expected value. The quality of such approximation may be found e.g. using the standard deviation of $\tau_j$. Please, tell me if you are interested in any further details.