As Nai said, with a two-hour window get a cab from the airport to the hotel (or to the hospital if you're really pushed for time). Not worth the aggro of stuffing around with skybus and trams to save, I dunno, $20 once you've factored in skybus tickets and myki tickets. I know money is tight but I think for this one I'd definitely prioritise making it as stress-free as possible.

Also, it's worth weighing up the costs of an apartment with a kitchenette versus just a hotel room. I find I save heaps if I've at least got somewhere I can make a bowl of pasta and a simple sauce versus getting takeaway or eating out for dinner.