How can I test for an expected exception with a specific exception message from a resource file in Visual Studio Test?

I would recommend using a helper method instead of an attribute. Something like this:

public static class ExceptionAssert
{
  public static T Throws<T>(Action action) where T : Exception
  {
    try
    {
      action();
    }
    catch (T ex)
    {
      return ex;
    }
    Assert.Fail("Exception of type {0} should be thrown.", typeof(T));

    //  The compiler doesn't know that Assert.Fail
    //  will always throw an exception
    return null;
  }
}

Then you can write your test something like this:

[TestMethod]
public void GetOrganisation_MultipleOrganisations_ThrowsException()
{
  OrganizationList organizations = new Organizations();
  organizations.Add(new Organization());
  organizations.Add(new Organization());

  var ex = ExceptionAssert.Throws<CriticalException>(
              () => organizations.GetOrganization());
  Assert.AreEqual(MyRes.MultipleOrganisationsNotAllowed, ex.Message);
}

This also has the benefit that it verifies that the exception is thrown on the line you were expecting it to be thrown instead of anywhere in your test method.