We study the first-passage properties of a jump process with constant drift where jump amplitudes and inter-arrival times follow arbitrary light-tailed distributions with smooth densities. Using a mapping to an effective discrete-time random walk, we identify three regimes determined by the drift strength: survival (weak drift), absorption (strong drift), and critical. We derive explicit expressions for exponential decay rates in the survival and absorption regimes, and characterize algebraic decay at the critical point. We also obtain asymptotic behavior of the mean first-passage time, number of jumps, and their variances for processes starting either close to the origin or far from it.