Running Docker on a cloud dev machine, with root on a real VM

You have root on a real Linux VM. That single fact is why Docker behaves here the way it does on any server you own, rather than the way it does inside someone else's sandbox.

Root means Docker

Pull an image and run it, no special mode and no permission dance:

docker run --rm -it -p 8080:8080 my-service

Bind mounts, custom networks, and compose files all work, because the daemon is running on your machine and you own it.

Reaching a container in a browser

A service you publish on a port is listening on the machine. From the machine's own terminal or editor you reach it at localhost:8080. To open it to the wider internet you point a domain at the box and put it behind a certificate — the same as you would on any VPS.

A note on room

Images and volumes use disk, and layers add up faster than people expect. If you build a lot, keep an eye on docker system df and prune what you are not using — or pick a size with more disk from the start.