Get it here:
https://www.photo.gallery/downloads/
Do I need to re-authorize the instance during the upgrade procedure?
No need. X2 admin does not require auth any longer.
If so: Unfortunately I've seemingly lost the Order ID. Can you somehow provide it to me (given some details like FQDN, Registration E-Mail, etc.)?
If you can provide the email attached to your license.
FYI: You are possibly aware (at least you should be) that X2 is very old and outdated by now. Even Google Chrome browser does not ship with Flash player since recently, which means even most desktop visitors will not see X2 properly (just a html fallback). I am curious, if you are spending time, why not use X3 to create a fast modern website? You can easily recreate the left sidebar in X2 (for example
http://pieterprins.nl/, more
https://www.photo.gallery/showcase/).