Converted this to a discussion.

Do you know what service is running on port 8080? Maybe code-server was added as a service and started automatically, removing the need to start it manually.

If not, I would say to simply launch code-server with a different port with

code-server --bind-addr 127.0.0.1:8081

and edit the Caddy config accordingly. Sadly, we can't help with installing Caddy on a server.

However, if you're using a LAMP server, I wouldn't recommend an additional Caddy server. Two options

1. Start code-server --link and use our tunnel. No need to reverse proxy or set up domains. This would also fix your port issue.
2. Use apache2 proxypass to proxy code-server. Here is the unofficial config but it hasn't been tested in a while. Make sure to change from 8080 to 8081 if you end up changing the port.