A quick suggestion, because it at least seems to have worked for me. Try to apply the 1.02 patch. When it fails due to the game not being there (you'll understand what I mean) open the patch using something like 7zip. (If you have it, right click on the patch in Windows Explorer, then go to the 7zip context menu and click on "Open Archive". From there go to $[78] and copy MassEffectLauncher.exe to the root of your Mass Effect directory (Likely something like "C:\program files (x86)\Origin\something\Mass Effect")
Then try running the patch again.
I honestly don't know if it will work for you. It did for me, or something happened behind the scenes that I don't understand. I've finally got the game working and I really don't want to screw around with it to figure out exactly which step did the trick. It might be you have to run the launcher once and then try the patch.
Oh, there's a config utility that will say you've got the current patch. You don't. It lies.
