Access to healthcare services is vital for urban public health, significantly influencing life expectancy, equity, and economic development. This study investigates the spatial accessibility of healthcare services in Rome through public transportation, emphasizing system resilience during disruptions. We explore transport poverty, emphasizing that individuals without cars often rely solely on public transit for medical access. Our methodology integrates network analysis with spatial accessibility assessments. We constructed a graph model of Rome's transit system, where nodes represent stops and edges connect neighborhoods to healthcare facilities. This model incorporates actual travel times from the city's official public transport timetables. We applied centrality metrics to identify crucial transit hubs and evaluated how their removal impacts travel times to healthcare facilities. The findings reveal significant disparities in accessibility resilience, influenced primarily by network redundancy and the strategic importance of high-centrality nodes. To enhance resilience, it is essential to monitor critical transit nodes and implement real-time flow monitoring to respond to disruptions. Collaboration among local authorities, transport agencies, and healthcare providers is crucial for risk assessment and identifying vulnerable populations. Developing targeted interventions and strengthening network redundancy will ensure more equitable and reliable access to healthcare, particularly for those dependent on public transportation.