We consider a stationary queueing process $Q_X$ fed by a centered Gaussian process $X$ with stationary increments and variance function satisfying classical regularity conditions. A criterion when, for a given function $f$, $\mathbb P (Q_{X}(t) > f(t)\, \text{ i.o.})$ equals 0 or 1 is provided. Furthermore, an Erdös-Révész type law of the iterated logarithm is proven for the last passage time $ξ(t) = \sup\{s:0\le s\le t, Q_{X}(s)\ge f(s)\}$. Both of these findings extend previously known results that were only available for the case when $X$ is a fractional Brownian motion.