The patcher needs ie6 installed to work properly. As CXGames does not have an entry for installing IE6 (CXOffice does), you'll need to use winetricks to install it.
1. Create a bottle for Anarchy Online (ie. "ao" - or just use your existing install)
2. Download winetricks (wget winezeug.googlecode.com/svn/trunk/winetricks)
3. Run the following commands (note that these paths will most likely differ on Mac, adjust accordingly):
export WINEPREFIX=~/.cxgames/NAME_OF_YOUR_ANARCHY_ONLINE_BOTTLE
# The path may differ if you installed using the .run packages
export WINE=/opt/cxgames/bin/wine
# This will install dependencies for freerealms
sh winetricks ie6
# Follow the wizard to install ie6
4. Once it finishes, you can launch Anarchy Online as you usually would, and the patcher will work.