There could be some useful functionality that could be found in startup and health probe topics [1] in Kubernetes for example. But at least in that world, it may be easier, and more transparent, to plug in a startup bash script to run before the main app. For example the Postgres Docker image [2] will run any scripts that may be mounted in a docker-entrypoint-initdb.d directory before starting the server. So putting a bash script in a ConfigMap that gets used as a disk volume, could be easier that downloading/auditing a binary "wait for it" container image.
[1] https://kubernetes.io/docs/tasks/configure-pod-container/con...