I had the pleasure to stay one month at Harry's Apartment while I was working at Monash University. The Apartment is just across Albert Park and offers a lovely view on the city center (CBD) from its large deck. The private room is simple, however, I have stayed most of the time in the upper floor where a large open living room with kitchen awaits you.
Harry was very welcoming and bought alot of typical australian food to try, and also invited me to join to have a drink in a bar on multiple occassions.
You can get to the Center in 20min with multiple trains or buses, and the Beach is a 10min walk down St Kilda Road. You can also enjoy your time at St Kilda Road at many Restaurants and Bars.