Adobe Illustrator CS4 does not let you directly apply special styling such as a gradient on the stroke of a shape but there is a workaround. This post walks you through the process to get it done.
Step 2 – Add a stroke to the shape by choosing Effect – Path – Outline Stroke.
Step 3 – The stroke appears with a solid color.
Step 4 – Convert the solid stroke into a separate object by choosing Object – Path – Outline Stroke.
Step 5 – You now have separate objects so you can select the stroke separately to make changes.
Step 6 – Finally, you can apply a gradient style on the separated stroke object and can apply other effects like a drop shadow. Rearrange the order of the 2 objects so that the drop shadow of the stroke does not cover up the inner shape.