Rhino V4 was designed to run under Windows XP, but it can run as originally designed in modern versions of Windows. Here are the limitations and requirements to run Rhino V4 under newer versions of Windows.
In addition to the new features and capabilities in Rhino V6, to take full advantage of the features of your new 64-bit Windows system, including RAM, display resolution, etc., you should consider upgrading to Rhino V6. Back in January 2007 when Rhino V4 was released, it supported the features of 32-bit Windows XP. Windows and computer hardware have come a long way since then. To take advantage of these new features and capabilities, you will need a version of Rhino designed for it.
Try the free 90-day evaluation license for Rhino V6 and see if you like it better on your new computer.