U can copy/past to new pc just fine. (Don't drop it in program files but in C:\UT2004 for example)
After that you need to get the CD key from the old computer to "register" it on the new PC.
at old computer:
* Run regedit and go to "HKEY_LOCAL_MACHINE \ SOFTWARE \ Wow6432Node \ Unreal Technology" or search for "CDKey"
* Right click on "Unreal Technology" name and select Export, give it a name and location and click save. (this file contains your UT2004 CDKEY, view/modify the file to check it contains the CDKey)
then you can:
a) run the reg file on the new pc to register ut again (if the old PC has a 32bit windows installation skip this part) or
b) run
UT2004RegFix-1.3.zip to do it manually.