Check out Watchtower! Auto-update your containers. Don’t forget to set WATCHTOWER_CLEANUP to true, or your disk will be filled with old images.

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:

https://www.cherryservers.com/blog/how-to-update-docker-image

You could use a systemd unit file:

[Unit]
Description=docker_compose_systemd-sonarr
After=docker.service 
Requires=docker.service

[Service]
TimeoutStartSec=0

WorkingDirectory=/var/lib/sonarr

ExecStartPre=-/usr/bin/docker compose kill --remove-orphans
ExecStartPre=-/usr/bin/docker compose down --remove-orphans
ExecStartPre=-/usr/bin/docker compose rm -f -s -v
ExecStartPre=-/usr/bin/docker compose pull
ExecStart=/usr/bin/docker compose up

Restart=always
RestartSec=30

[Install]
WantedBy=multi-user.target

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.

no need for docker compose down. pull && up is enough

Also depends on how you specified image in the docker. If it has no version or latest as version it will update otherwise it may be fixed

If you set up using compose and don't have the version pinned:

dockee compose down && docker compose pull jellyfin && docker compose up -d