CNAME with external domain via Tunnel and domain on CF

Greetings,

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:

Setup:

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 sub.example.com
The container has no published ports and the Cloudflare Tunnel is exposing container01.example.cloud

What I now need (and what I tried without success) is the following:
CNAME sub.example.com pointing to container01.example.cloud 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 sub.example.com is not on cloudflare sadly. Which is the cname reason to my domain on cloudflare.

Any ideas if this could work on CF?
Thanks!