Hi, Cornman21,
A radar dish like that is actually a parabollic shape rather than a cone shape. If you go online, you can find a lot of sites that provide instructions on how to draw a parabolloic curve, or you may be able to find some cross section drawings you can use to make a template. Make two curves - one more pronounced toward the center for the inside of the supports, and one more shallow for the outside of the supports.The supports should be thicker at the center and thin at the tips. You'll have to experiment.
You can make the antenna supports from single strips of styrene (5 inches long in this case). There are usually not a lot of supports (10 is probably enough) because the actual structure had to fold up for launch.
I would make a circular disk about 1/2 inch in diameter for the inside of the dish and glue your supports to it with them radiating out from the center.
I don't know about the Casini antenna, but the surface of many space antennas is just a thin metallic-coated (usually gold) Mylar material. You could use clear food wrap glued to the inside of the supports for this. To get the metallic sheen, you could spray a thin layer of gold spray paint over it.
As a last ditch effort, there are a lot of Lego space-related sets that have parabolloid antennas... ;)
Good luck!
Russ