Make sure that you are doing the Update with antivirus and firewall(s) disabled.
Please read the following for solutions when problems exist with that 13.1 Update.
When you get the 13.1 Update, you should expect the following
Please let us know the outcome as your schedule permits. Any questions or need clarification, please do not hesitate to ask.
At this point, I'm not sure if the update was already complete or not. I went into Photoshop Elements, clicked on update. I received instructions to close Premier Elements. I followed the instructions and voila both the Photoshop and Premier successfully were updated to 13.1. So I guess we can mark the question as answered.
Thanks for the follow up.
At this point, if you are you getting 13.1 listed when you look at
Premiere Elements 13 Editor/Help Menu/About Adobe Premiere Elements
Photoshop Elements 13 Editor/Help Menu/About Photoshop Elements
Elements Organizer 13/Help Menu/About Elements Organizer
(whether you got to the Elements Organizer from the Photoshop Elements or Premiere Elements side of things).
Then the updates are done. Great news. Great job sorting through and staying with the task.