To answer the question at hand; absolutely, non-residents can buy shares in Australian listed company
Most brokers in Australia require you to submit your tax file number (TFN)...so unless you have an Australain TFN (which I assume you don't) you will most likely have to go through a respected broker in Ireland/most banks around the world should have this facility available to you.
So in short...join a local brokerage firm/walk into a major bank and from there it shouldn't be to hard to invest in Australian shares.
Just looking at my broker, they offer me the ability to invest in Irish shares with a click of the button.
I can't speak for online brokers, but if you use a full service broker, many do not even require a TFN, and you can use Ireland and your address, it does not have to be in Australia.
The hardest part will be getting your ID certified by an acceptable person to meet our AML Laws as you cant just walk in to any Australia Post!
Yes, theasxgorrilla is correct. You can set up on interactive brokers a/c from Ireland (I did) and trade stocks on any exchange. You will need to pay for market data if you require it however.