exactly what it says in the title.

But... update the game itself? It should do so automatically, if there is a new patch/version avaible. But, if you want to check if everything is okay:
1. On left panel with all your games right-click on AoW3
2. Go to Manage Installation
3. Click on Verify/Repair

This will check if there are newer or different versions of the game on server and download the differences.

Verify/Repair is greyed out. The only options not greyed out there are Show Folder and Uninstall.
I had problems installing the DLCs, only one of them detected the installation directory of the game truely which resides on D: drive, the other 3 insisted on installing to C: drive(the installers giving no option to change directory at all).
Probably the cause of those options greying out ?

Seems so... though this is definetly odd. You can set the install destination for the games in general settings... but it applies to all of the games. Probably at this point you should uninstall it and install it again... though maybe GOG's tech support could help you better? Honestly, I'm not sure what else (aside unistall) to do in this case...