Given a Gaussian random walk $X$ with positive drift, we consider estimating its
first-passage time $\tau$ of a given level $l\geq 0$, with a stopping time $\eta$
defined over an observation process $Y$ that is either a noisy version of $X$,
or a delayed version of $X$. For both cases, we first provide lower bounds on
the mean ${\mathbf{E}} |\eta-\tau|^p$, $p\geq 1$, for any stopping rule $\eta$.
Then, we exhibit simple stopping rules that achieve these bounds in the large
threshold regime and in the large threshold large delay regime, respectively.