CNAME with external domain via Tunnel and domain on CF


I have a question. I have a few docker containers that are only accessible via the tunnel.
The containers are a own creation and are running nginx as well.

I have the following requirements and wanted to check if this is possible with the tunnel, as I can’t get it to work:


Docker host running a container of cloudflared.
Docker host has a few of those special docker containers as well.

Special Container 01 has a set server_name of
The container has no published ports and the Cloudflare Tunnel is exposing

What I now need (and what I tried without success) is the following:
CNAME pointing to and displaying the website of the docker container as the server_name matches.

What actually happens:
The tunnel reports back the catch-all 404 status code. And i’m not sure if I could get this to work.
The domain is not on cloudflare sadly. Which is the cname reason to my domain on cloudflare.

Any ideas if this could work on CF?