We investigate the first-passage properties of a jump process with a constant drift, focusing on two key observables: the first-passage time $τ$ and the number of jumps $n$ before the first-passage event. By mapping the problem onto an effective discrete-time random walk, we derive an exact expression for the Laplace transform of the joint distribution of $τ$ and $n$ using the generalized Pollaczek-Spitzer formula. This result is then used to analyze the first-passage properties for two exactly solvable cases: (i) both the inter-jump intervals and jump amplitudes are exponentially distributed, and (ii) the inter-jump intervals are exponentially distributed while all jumps have the same fixed amplitude. We show the existence of two distinct regimes governed by the strength of the drift: (i) a survival regime, where the process remains positive indefinitely with finite probability; (ii) an absorption regime, where the first-passage eventually occurs; and (iii) a critical point at the boundary between these two phases. We characterize the asymptotic behavior of survival probabilities in each regime: they decay exponentially to a constant in the survival regime, vanish exponentially fast in the absorption regime, and exhibit power-law decay at the critical point. Furthermore, in the absorption regime, we derive large deviation forms for the marginal distributions of $τ$ and n. The analytical predictions are validated through extensive numerical simulations.