Do AutoCAD 2012 customized menus work in AutoCAD 2013 software?

Yes, I do not think there was a major revision to how the .cuix files are defined and used.  There were changes several years ago when .mnu files became .cui and then .cui became .cuix.  I believe the menu file definition is now pretty stable, and you won't have problems.

 

I have not tried it in AutoCad, It looks like the the2 programs are very close together, therefore usually are compatible and you should not have problems properly import your menus. 

To be on the save side check with autodesk.com

 

 

knowledge.autodesk.com/.../autocad/.../Right-Click-menu-is-mi...   Autodesk Nov2,2014 - Applies to AutoCAD2012, AutoCAD2013, AutoCAD2014, AutoCAD ...Customization... button and check Shortcut Menu for Default Mode, Edit ...

How to migrate settings in AutoCAD - Autodesk Knowledge ...

knowledge.autodesk.com/.../autocad/.../How-to-migrate-setting...   Autodesk Apr24,2015 - After installing a new version of AutoCAD, you want to know how to migrate the ...2008, AutoCAD2009, AutoCAD2010, AutoCAD2011, AutoCAD2012,AutoCAD2013, ... option under the Migrate Custom Utilities menu to export those settings. ... Except where otherwise noted, work provided on Autodesk ...

Most of the menu's of Autocad Menu's should work in Auto-Cad2013.