I live in the area so can vouch for walking distances. The hospital borders Parkville and North Melbourne so search both those suburbs when looking around as North Melbourne might have more options

The Park Squires hotel in Flemington road is definitely walkable. Flemington road is a major road, very noisy but you would walk along side a park for the whole way with only one road to cross so it's a good choice. The only issue I have with that one is that it is at a pretty busy intersection, where flemington rd intersects with citylink, I would imagine it is pretty noisy. Another hotel close by is Rydges Nth Melbourne. Across flemington road and a short walk but it is close to public transport and closer to the local shops etc for a bit of Melbourne cafe lifestyle. If cost is a real issue, the Arden (already mention) is pretty cheap, I think about $80 night last time my folks stayed. It is a bit more of a walk but absolutely do-able. The quality can be a bit hit and miss from accounts I have had from family and friends but it is super close to town, the market etc. There are also serviced apartments around we have had family use via wotif at a reasonable rate, try the British Apartments on Arden St.

Check out the public transport victoria website for public transport planning. It is fab as you can enter your start and end point, your arrival or departure time and it will give you the best public transport options (train, bus, tram, walking) with maps. The hospital is on tram route 55 and 59.

As for getting from the airport: a taxi tends to cost us about $40-50 from the airport to parkville/north melbourne. Skybus is in the vicinity of $17 each with hotel transfers available. You otherwise have to get a Myki (full fare 3.50 - 2hrs, 7.00 - all day) and make your way via tram/train to your accommodation. Considering the cost difference, it might be worth your peace of mind to just get a taxi to accomodation, get settled and take your time to get to the appointment.