So I was having the same problem many others have been having. I would set for high res and my screen would be cut off once the game launched as it would revert back to low res.

Here's what I did to solve it.

Right click the Anachronox link and go to compatibility.
Check to run as admin.
Check the box to "disable display scaling on high DPI settings."

Boom, able to run in high res on full screen with no problem.

GOG.com\Anachronox\anoxdata\SAVE\GLOBAL\settings.cfg

The line you are looking for is - set u_gl_mode "9"
Set the number for the resolution you want.

0 for 320 x 240
1 for 400 x 300
2 for 512 x 384
3 for 640 x 480
4 for 800 x 600
5 for 960 x 720
6 for 1024 x 768
7 for 1152 x 864
8 for 1280 x 960
9 for 1600 x 1200


After that save the file and game starts already in the desired resolution. Number 6 worked for me fine.

My monitor is a 16:10 (1440x900 is the native resolution) so how can I get this game to scale up to that?

Right click, select Properties, then Compatibility tab, to find [x] Disable display scaling. I did not need this for the builtin 'hi res' (# 8 above) but did for custom resolution (#9, 1600x1200) I have a widescreen monitor also. For games like this, things get stretched horizontally. Pick the highest resolution your card and monitor support. Probably 7 for 1152 x 864, certainly 6 for 1024 x 768 should work.