The ArcObjects SDK 10.1.X only supports Visual Studio 2010. To utilize it in a supported manner you'd have to get one of the following versions of Visual Studio.
ArcObjects SDK 10.1.X system requirements
http://resources.arcgis.com/en/help/system-requirements/10.1/index.html#//01510000006n000000
- Microsoft Visual Studio 2010 (C++, C#, VB.NET) Professional, Premium, Ultimate Edition
- Microsoft Visual Studio 2010 Visual Basic Express
- Microsoft Visual Studio 2010 Visual C# Express
You could use 2013 to modify an existing ArcObjects solution or you can install the templates manually, but our software would not integrate with it. The ArcObjects 10.3.X install would support the use of Visual Studio 2013.
ArcObjects SDK 10.3.X system requirements
https://desktop.arcgis.com/en/desktop/latest/get-started/system-requirements/arcobjects-sdk-system-r...
- Microsoft Visual Studio 2012, 2013 (C++, C#, VB.NET) Professional, Premium, Ultimate Edition
- Microsoft Visual Studio 2012, 2013 Visual Basic Express
- Microsoft Visual Studio 2012, 2013 Visual C# Express
ArcObjects SDK 10.2.X system requirements
https://desktop.arcgis.com/en/desktop/latest/get-started/system-requirements/arcobjects-sdk-system-r...
- Microsoft Visual Studio 2010, 2012 (C++, C#, VB.NET) Professional, Premium, Ultimate Edition
- Microsoft Visual Studio 2010 Visual Basic Express
- Microsoft Visual Studio 2010 Visual C# Express
