Skybus do the hotel drop off (as Rouge said). If your hotel isn't in the drop off list, then the skybus will drop you off at Spencer Street (Southern Cross) station.

Honestly with a tired little man (after a flight) & luggage, I'd get a cab from the station to your hotel. You're looking at about $20 for the cab from the skybus stop to near the RCH.

Trams that go past the RCH are the 55, 57 and 59.

Look around Parkville and North Melbourne for accomodation. At a pinch you could stay on the border of Brunswick/Princess Hill (along the tram # 19 line, as the tram goes past the main hospital).

The hospital should be able to give you a list of suitable accomodation nearby.

You will need to get a myki for you both when you arrive so that you can travel on public transport. There is a 7-11 at Spencer Street that sells them (just near the skybus stop).