If you don’t need the full ASP.NET MVC 4 installer with project and item templates for Visual Studio 2010 then you can download the various ASP.NET Web API packages directly from NuGet (see list of packages below). If you are new to NuGet then it is a Visual Studio extension that makes it easy to add, remove, and update libraries and tools in Visual Studio projects that use the .NET Framework (check out the NuGet overview).
Here are the relevant NuGet packages depending on what you want to build:
Have fun!
Henrik