Service mobility in Multi-access Edge Computing (MEC) paradigm is necessary to provide ultra-Reliable Low Latency Communications for the erratically roaming MEC users. It involves relocation of containerized application services to a strategically selected optimal edge host. During relocation, service containers are unavailable (downtime), resulting in the interruption of ongoing user sessions and increased operational expenses for the network operator. Prolonged service downtime degrades perceived quality of experience for users, and this study handles this problem by proposing a downtime-aware Policy Learning based Capped Downtime (PLCD) service mobility strategy. It exploits Deep Actor-Critic prowess for effectively deciding when and where to relocate a containerized application service while taking user mobility and MEC server resource fluctuations into account. Efficacy of the proposed PLCD strategy is confirmed through simulation experiments, and results indicate over 90% average reduction in service downtime comparing to a baseline scheme.