Free internet is available in every McDonalds (if it works and is not overloaded). Anyways, if you need to work remotely we definitely suggest the State Library Western Australia in Perth.
You can also use one of their computers for 40 minutes per session. If all computers are occupied in the library, you can reserve a computer. You get the exact time when a computer will be free, so you can go and use it.
Generally speaking, libraries always offer free internet in Australia, with a few exceptions.
25 Francis St, Perth WA 6000, Australia-31.9489516115.8604006