SOLUTION: For me I simply unloaded the project, edited the project file to change this line:
<Import Project="$(MSBuildExtensionsPath32)\Microsoft\VisualStudio\v10.0\WebApplications\Microsoft.WebApplication.targets" Condition="false" />
to this:
<Import Project="$(MSBuildExtensionsPath32)\Microsoft\VisualStudio\v11.0\WebApplications\Microsoft.WebApplication.targets" Condition="false" />
After saving and reloading the project file the upgrade wizard automagically kicked in and did some magic and the publish started working.
Aftewards I went in and had a look at the project file again and it seemed the upgrade wizard added a PropertyGroup element with VisualStudioVersion and VSToolsParth sub-elements (similar to what this guy recommended manually adding but for me manually adding them didn't work - <Import Project="$(MSBuildExtensionsPath32)\Microsoft\VisualStudio\v10.0\WebApplications\Microsoft.WebApplication.targets" Condition="false" />)
You'll also find lots of links about renaming Microsoft.WebApplication.targets and running a repair (http://stackoverflow.com/questions/10989051/why-do-i-get-the-error-the-target-gatherallfilestopublish-does-not-exist) - which I tried but this didn't fix the issue.