How do you get notified when a long terminal command finishes?

Try a push notification service such as ntfy or Gotify. You can send a message from a shell command and receive it on your phone. That's a pretty direct fit if you don't want to set up email delivery.

For a backup, make the message distinguish success from failure. In a shell, you can use an if statement around the backup command and send a different notification in each branch.

Email is perfectly reasonable if the machine can already send it. Run the backup, then use a mail utility to send a short message. A semicolon between the commands runs the mail command whether the backup succeeds or fails, so capture the exit status if you want to report the result.

On the phone, a mail rule or priority sender setting can make those messages more noticeable. The awkward part is getting outgoing mail configured if it isn't already.

For a notification on the same Linux desktop, notify-send is a quick option. You can add it after a command, or wire it into an interactive shell so commands above a time threshold trigger a notification automatically.

The automatic route needs some filtering. Editors, pagers, and other programs you deliberately leave open can otherwise produce a lot of noise.

If you already use a messaging app, a bot posting to a private channel is another possibility. It takes more setup than appending a simple notification command, but it gives you a place to collect completion messages from different machines.

Keep an eye on volume, though. A notification for every routine job can become easy to ignore.

For the simplest local alert, append a command that prints the terminal bell character. It may make a sound depending on your terminal settings. It won't notify your phone, but it requires very little setup.