I currently have a Let's Encrypt SSL certificate in place on my site, and everything is working correctly. However, I am now wanting to install a dedicated GoDaddy certificate instead.
My server setup is Apache on Ubuntu. Is it possible to install the new certificate and keep the Let's Encrypt one registered as a fallback, or do I need to entirely revoke the Let's Encrypt certificate first?
No. Revoking should only be done if it is no longer safe to use it. If you don't want to use the LE cert anymore just start using your new one.
In short, yes
There are no limits/requirements that would prevent you from keeping it.
Yes, you can even keep both simultaneously. Apache will send whichever it likes more. (depending on the negotiated parameters of the tls session, and if the certificates are actually different)