Like @neoakris writes, you can achieve your goal by defining a cloud run startup probe. Starting with gcloud cli version 501.0.0, you can now (currently only on the beta track) define startup probes without having to manipulate a service.yml directly with the gcloud beta run deploy command, like:

gcloud beta run deploy my-service --startup-probe=httpGet.port=8080,httpGet.path=/ready,initialDelaySeconds=10,periodSeconds=20,timeoutSeconds=20

See https://cloud.google.com/sdk/gcloud/reference/beta/run/deploy?hl=en#--startup-probe