Neither resolution nor the cheat menu graphics fix remain as I set them when I quit the game. Running the Launch Populous - The Beginning shortcut as admin. Obviously I can set them every time but that's not perfect.

Any known fix?

I know this is a very old topic, but if anyone has the same problem, this tool can change the resolution up to a maximum of 1600x900 and make the game remember it: https://www.popre.net/downloads.php?f=10

Note that it will still say 640x480 in the options menu and it will revert to this if you open the options from within the game. That is, until you returned to the main screen and reloaded the game, I believe.

It does not work for the Add-On, though.

Use dgVoodoo2 to fix graphics.

I do, but it can't change the game's resolution... at least not for me.

I was addressing OP's other problem.