You'd place your compose file in the working dir /var/lib/sonarr. Depending on what tag you've set for the image in the compose file, it would be autoupdated, or stay fixed. E.g. lscr.io/linuxserver/sonarr:latest would get autoupdated whereas lscr.io/linuxserver/sonarr:4.0.10 would keep the container at version 4.0.10. If you want to update from 4.0.10, you'd have to change it in the compose file.
Did you use docker compose file or just run a command to start the container?
Edit: I always use compose files. For that you can do the following:
docker compose pull
docker compose down
docker compose up -d
You don't technically need the stop, but I've found once or twice in the past where it was good to stop because of image dependencies that I forgot to put in my compose.
For running a command directly I found this website that seems to summarize it pretty well I think:
I see someone mention watchtower, while not a bad thing, I just prefer to manually update. This helps to ensure any breaking changes don't break my system. Especially with something like Immich at it's had a lot of them recently as they work towards stable. I just generally subscribe to their release and do updates as necessary.
Podman supports auto updating natively by setting a label.
I use systemd service files for running containers, but you can add the same label on the command line or in quadlet files.