Being on a working visa I would assume you would qualify for medicare which means you could give birth in a public hospital.
You would have to attended your appointments at the hospital, I don't think they have appointment times, but general clinic hours, so first in first served.
Hospitals won't force you into having interventions you don't want, so a natural birth is possible, but having the choice if you want pain relief is an option. If the bub is in distress or you labour doesn't progress/start they may suggest a number of different interventions such as CS, inductions, syntocin drips, forceps/vacuumes extrations, etc...
You are generally encouraged to labour at home untill you find the pain unbearable or your contractions are 5 mins apart, public hospitals can be busy. You will probably be d/c home within a day or so if there are no complications, possibly less than 24hrs. Then there will be follow up visits with midwives from the hospital and community nurses.
I have had birthed in a private hospital, so if my information is incorrect, someone please correct me. Since you don't have health insurance, it would cost a small fortune as you would have to pay for the appointments with you Ob in the lead up to labour, then you would have to pay for the hospital stay, blood tests, medications etc... Even with insurance the out of pocket expensises could be up to a couple of thousand dollars.

I hope some of this information has helped clarify things for you.