For some reason shipping from the US to Australia is absurdly expensive. I learned this a year ago when I tried to ship a tenor. Today, in response to this inquiry i went to a few sites where you can calculate shipping rates. I gave the weight and dimensions of a hypothetical soprano uke and the cheapest estimate I got was a little over $300.
There are however, ways to consolidate shipping. Businesses can do this and get cheaper rates. There are also several services, an example is Parcel It On. If you live in Australia you can arrange with them to give you a US address. A package can be shipped to you at the US address, they will then forward it to you in Australia. They consolidate shipping so their rates are dramatically lower than an individual can get. It costs considerably more to ship from California to Australia than it costs to ship to Europe. I don't know why but it does.