We consider ordinary and conditional first passage times in a general birth-death process. Under existence conditions, we derive closed-form expressions for the $k$th order moment of the defined random variables, $k\geq1$. We also give an explicit condition for a birth-death process to be ergodic degree 3. To the best of our knowledge, the results of this paper are new, except for some already known quantities as we shall mention. Based on the obtained results, we next analyze some interesting applications for markovian queueing systems. In particular, we compute for some non-standard markovian queues, the moments of the busy period duration, the busy cycle duration, and the state-dependent waiting time in queue.