Hello, and welcome to the forum. We are in Melbourne, so not too different from your climate. BTW, my eldest daughter just got back from a school camp in Canberra, and she loved it! I haven't tried
growing finger limes, but the Burke's Backyard website says they are worth a shot in cooler climates. Just mentioned the one specific variety at
http://www.burkesbackyard.com.au/1999/archives/25/in_the_garden/herbs,_fruit_and_vegetables/limes
We have 2 citrus trees, a lemon and an orange, and they are at the back of the garden, facing north. The lemon especially is doing well. I think the key is just to protect them from frost while young.