Catch a skybus or similar to your accommodation. I'd try and get accommodation close to the hospital. Public transport here is great. It probably will be cold and quite possibly wet. Check out stayz, wotif and last minute. I'd even call the hospital and see what accom they recommend to family who need it. They might know something awesome.

I am not sure a 2 he window is possible unless you have accom close to hossy. Check out the metlink website to plan public transport from where you decide. You might be worth going straight to RCH and then public transport to accom after. Keep in mind your appt could be up to 2 hrs late depending on the staff that day.