This paper studies a continuous-time joint sampling-and-preemption problem, incorporating sampling and preemption penalties under general service-time distributions. We formulate the system as an impulse-controlled piecewise-deterministic Markov process (PDMP) and derive coupled integral average-cost optimality equations via the dynamic programming principle, thereby avoiding the smoothness assumptions typically required for an average-cost Hamilton-Jacobi-Bellman quasi-variational inequality (HJB-QVI) characterization. A key invariance in the busy phase collapses the dynamics onto a one-dimensional busy-start boundary, reducing preemption control to an optimal stopping problem. Building on this structure, we develop an efficient policy iteration algorithm with heavy-tail acceleration, employing a hybrid (uniform/log-spaced) action grid and a far-field linear closure. Simulations under Pareto and log-normal service times demonstrate substantial improvements over AoI-optimal non-preemptive sampling and zero-wait baselines, achieving up to a 30x reduction in average cost in heavy-tailed regimes. Finally, simulations uncover a counterintuitive insight: under preemption, delay variance, despite typically being a liability, can become a strategic advantage for information freshness.