I have an outlook plugin built with Visual Studio 2010 (.net 4.0 Client Profile), that works fine in Outlook 2007 and Outlook 2010. I just tested on 2013 and even installing the addin seems to crash. After googling, I see a few examples of issues people have faced but fundamentally can't get a straight answer to this question:
Is there any prerequisite that would not allow a VS 2010 Outlook addin to just work on 2013 or do I need to upgrade Visual Studio and Create an Outlook 2013 specific version of my addin?
Do you have Outlook 2013 on the machine you have the add-in project on? If you do, you can debug from VS 2010 by setting the "Start external program" in the Debug tab of the project properties window to the Outlook 2013 .exe. When I created my add-in I was able to debug it using whatever version of Outlook was on my development computer and I don't remember changing anything other than the external program path.
If you are using setup project for the installation it needs a different ComponenedId when you create its launch conditions. You'll need two different setup projects to install on Outlook 2010 and 2013. Here are the Office 2013 PIA keys and Office 2010 keys.