You could use Tableau Bridge to avoid having to make your Aurora cluster public:
Bridge would connect to Aurora from your local network, and upload the data to Online securely.
Yes, but this still requires us to have one computer turned on 24/7 in the office. We were hoping to connect Tableau Online directly to Aurroa..
Ah, that is true, you would have to have a computer turned on to run this.
If you're looking to connect Tableau Online directly, then you would need to either put a proxy server in front of Aurora for Tableau Online to connect to (not sure if this is possible, you'd probably want to reach out to AWS on this) or make the Aurora node publicly accessible so Tableau Online could reach it, which lands you back at your original question.
If security is what you're most concerned with, I'd stand up a small VM on your local network, and use it to run Tableau Bridge, leaving it up and running to facilitate the connection. Or, use extracts and turn on the computer with Tableau Bridge to refresh them at an interval of your choosing.