It is as simple as tagging the Docker with gcr.io/$PROJECT_ID/$CONTAINER_NAME.

I prefer to create .env with the following lines,

PROJECT_ID=
CONTAINER_NAME=

Then, run source .env, or use Oh-My-Zsh with dotenv plugin. Then run,

docker build -t gcr.io/$PROJECT_ID/$CONTAINER_NAME .
docker push gcr.io/$PROJECT_ID/$CONTAINER_NAME

With Domain Mapping (Beta), it is dead simple to use a custom domain.

However, today I try Google App Engine (for Nuxt). I find it not as transparent as GCR, but not really harder...

Also, not sure about the costs.

There is this diagram, though. -- https://cloud.google.com/serverless-options