I'm no hat expert, but to me it simply appears to be based off of one of many Western style hats, but not the classic Stetson 'cowboy' type hat. I have no idea what the correct name for that style of hat it, if there even is one. However, if you browse through just about any website selling Western wear, you will find similar hats.
ETA: I assume they called it "Duster", since a duster was/is a type of Western-wear coat.
http://en.wikipedia.org/wiki/Duster_%28clothing%29
It's actually a "slouch hat," a type of hat commonly worn by US Cavalry troops during the Civil War and the later Indian Wars. Popular in the American Old West, too. (There's also a Wikipedia article about them that leaves out 1800s use in the States, but goes into considerable detail about the hat's usage history in British Empire countries -- the archtypal Aussie bush hat is a type of slouch hat, for example.)
Perhaps the "duster" name came from associating the slouch hat with the "dusters" -- the long trenchcoat-like coats -- worn in Spagetti Western and Peckinpah movies. (See the Wild Bunch as protrayed in MY NAME IS NOBODY, for example.)
The Shadow, Golden Age pulp magazine hero of the 1930s and 1940s, wore a slouch hat. It's always described as by that name in the texts.
Hope this helps.
-- Doc Hornet
Thank you Hornet! That was exactly what I wanted.
I like the hat called "Duster," but I can't find any evidence that any type of headgear has ever been called a "duster" in real life. What would that hat be called in real life?
(I browsed some "hat type" galleries, but the distinctions between various hats seem to be pretty subtle at times.)