I bought PoE back after it first came out via GoG and I wanted to go back and finish the game, however after checking for newest patches, it seems to be that I am missing one patch.

On the main menu, lower left, it says version 2.02. However patches on GoG start at 2.03 if I'm reading it correctly and when I try to install that it says I have the wrong version.

So how do I patch the game?

If you don't pay your connection by megabyte, it's best you backup your saves, uninstall current version and install the latest version via full installer from your library. Savegames are compatible and you can restore them and continue the game.
I like GoG because of the DRM-free policy, but patching here is a big mess.

Also bear in mind that version 3.04 (fixing some critical bugs) went live few days ago and it will appear here in a day or two. So you may want to wait a bit more and install full 3.04 game.

Yeah, wait for patch 3.04 to go live here, then uninstall/reinstall with full installers. Much cleaner.