Cloud Run uses Docker containers, but it does not support Docker Compose. If you're relying on that docker compose yaml to wire up your containers, it's not going to work. If you want to use a yaml file to store your service configuration, Cloud Run uses a Knative service as its native Yaml format.

Cloud Run also doesn't support disks, so unless you want to have a very slow database serving off of a GCS storage bucket or a very expensive database serving off of a $400/month NFS share, you probably should run that in Cloud SQL and use the built in connector to connect the database to your Cloud Run Service.