Firstly. there's an official patch for this game (https://legacy.3drealms.com/dukemp/patch.htm) so why does Gog Galaxy says it's only v1.0?
EDIT: I did as 3DRealms instructed and opened the console. It says it's v1.01.

Secondly, when I tried to install the patch, it says the game is not installed and fails to patch.

If anyone has a way to manually patch the game, that would be appreciated.

Version numbers shown in Galaxy are typed in manually, they are not necessarily the same as the dev used for the game.

If the console says 1.01, then you don't have to patch.

But if you insist, you an always try to copy the patch file to the game's directory and run it from there. Some patches need that.