Enter a docker container running with Google Cloud Run

Yes you can, with a slight hack - I've released an example here https://github.com/matti/google-cloud-run-shell

You just need to copy these binaries over to your container and establish the reverse shell access through heroku or something similar.

Is it possible to enter a container powered by Google Cloud Run?

No, the only port you can access the service is $PORT (currently 8080) and only via HTTP/HTTPS requests.

You will need to debug the old fashioned way: logging statements to Google Stackdriver - which is already built-in to Cloud Run, just do "logging.info(...)" for your language.

If your problem is container startup, check Stackdriver for error messages.