Before you install Opera Mail on the new computer:
Put "C:\Documents and Settings\user\Application Data\Opera Mail" in "C:\Users\user\AppData\Roaming".
Put "C:\Documents and Settings\user\Local Settings\Application Data\Opera Mail" in "C:\Users\user\AppData\Local".
Inside "C:\Users\user\AppData\Roaming\Opera Mail" and "C:\Users\user\AppData\Local\Opera Mail", there should be folders named "Opera Mail" to match the name of the "Opera Mail" program files folder. If they're not named "Opera Mail", rename them. (Don't rename the parent "Opera Mail" folders though).
Also, edit "C:\Users\user\AppData\Roaming\Opera Mail\Opera Mail\operaprefs.ini" with a text editor. Remove any line that has a path in it so that Opera Mail uses the default location for that setting.
Then, launch the Opera Mail installer, click "options" and set the "install path" to the program files folder. For 64-bit Windows 8, set it to "C:\Program Files (x86)\Opera Mail". For 32-bit Windows 8, set the "install path" to "C:\Program Files\Opera Mail".
Then, set "install for" to "All Users", set your icon options and install.
Done properly, when you first launch Opera Mail, it'll be just like Opera Mail was always on windows 8.
To make Opera Mail the default client, see this zip file for a reg file that you can import.