I'm fairly majorly arachnophobic and grew up in Australia. You will be fine.
In newer flats and hotels you really won't see them. You get spiders mostly indoors only in older houses that aren't well fly-screened and/or have trees/bushes/plants right next to the windows - that's how they get in.
The only place you might see them out and about is if you were to use an outdoor public toilet type place - eg changing rooms at the beach, they may be hiding in the ceiling/eaves but you don't see them much indoors. Just don't leave your shoes outside!
I think it's very likely you could spend 2/3 weeks in Aus on holiday and never see one up close