I took a different approach, Setup your nginx or http server to start up immediately as soon as your preparing steps take place, point to the same port I just made a dummy index.html has one line

Site is loading...please wait

at least it returns 200OK response to satisfy the probing requirements and it won't fail. then after your know your site is up and ready , or slightly before shutdown or restart nginx or http to point to your site. Unfortunately, the Cloud Run has one single use case to deploy and run dynamic content with very limited configuration. Wish you good luck, hope this solution works!