Apache Zeppelin is using Micrometer - a vendor-neutral application metrics facade.
Prometheus is the leading monitoring solution for Kubernetes. The Prometheus endpoint can be activated with the configuration property zeppelin.metric.enable.prometheus. The metrics are accessible via the unauthenticated endpoint /metrics. For Grafana a good starting point for a dashboard can be found in our Github Repository.
JMX is a general solution for monitoring Java applications. JMX can be activated with the configuration property zeppelin.jmx.enable. The default port 9996 can be changed with the configuration property zeppelin.jmx.port.
Apache Zeppelin has two healthcheck related unauthenticated endpoints (/health/readiness, /health/liveness) that could be used for proxy and/or cloud setups.