This is not so much a question as a potential answer for the edification of others regarding this error 0x800F0A12 that I got several times when trying to install Windows 7 Service Pack 1. Maybe this can help someone else with the same or similar problem.
In my case, the error was being caused by the fact that I have Windows 7 Professional installed on a Hybrid GPT/MBR disk. Windows 7 32-bit likes an MBR partition and my other OS prefers a GPT partition for maximum disk performance. The other OS is not important to the error message at hand, so no need to mention it here.
Additionally, I was using a third-party boot manager with the active partition being the partition with another OS that runs on a GPT partition. Apparently, the installer for Windows 7 Service Pack 1 looks to the active partition for the boot files that it needs to update.
If the Windows 7 partition is not the active partition the updater does not know where to find the files it needs to update and generates error 0x800F0A12. There was and is no System Reserved partition on my hard disk, either.
The problem was solved immediately when I set the Windows 7 partition as the active partition. After doing that, installation of Service Pack 1 went and completed without a hitch.
Here is what I did. I cannot guarantee that this will work for you but it did for me.
Before running Service Pack 1 installation, check to be sure that your Windows partition is marked as the active partition. You should do this from within Windows 7. I used two programs to do this and to change it back. If it already is the active partition what I did and wrote below will not help you.
First, I ran Disk Management from within the Computer Management console. I then right-clicked the Windows partition and marked it active. I saved changes and exited the Computer Management Console
Then, I ran the Windows 7 x86 Service Pack 1 installation from a DVD that I created from the ISO file supplied by Microsoft. I see no reason why Windows Update will not work as well, but if it doesn't, download the ISO directly from Microsoft.
You will need to Validate your Windows 7 installation to download the file. Do not risk getting it from somewhere else or you could make things worse for your system.
After the system restarts and finishes the update and configuration, run DISKPART from the Command Prompt as an Administrative or as the Administrator user and go through the steps to mark the partition from which your third-party boot manager runs as active (if that is your configuration as it is with my system). You will not be able to do this from within the Computer Management Console again so DISKPART is what you need.
Finally, after working with Microsoft's DISKPART, restart your system again and everything should be working as it should. If not, you might need to reinstall or reactivate your boot manager again.
Note: If you messed up your hard disk by running any other commands to fix the MBR or to fix boot code for Windows 7 you may have further problems that this information cannot help.