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.
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.
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.