Good luck getting to Phillip Island. Not too much on this Sunday off the top of my head.
There is the Melbourne International Film Festival, but I don't think you want to see a movie as part of your 1 day here.
Richmond & Essendon are playing at the MCG, if you wanted to see a game of Aussie Rules Football. First bounce at 2:10, and would finish at approx 4:50.
Regarding food/drink, it depends on what you are after but I wouldn't put any Schezuan place on the top of my list. But dinner is hard on a Sunday night when you don't have a booking.
As above I would recomment St Ali (12-18 Yarra Place, South Melbourne) for breakfast & coffee. Other options are Dench bakery (109 Scotchmer St, Fitzroy North), Richmond Hill Cafe & Larder (48-50 Bridge Rd, Richmond) and Mr Tulk (State Library, 328 Swanston St, Melbourne CBD). Just depends on where you are in and around the city.
Lunch? Take your pick from the resturants at Crown Casino.
If you have some spare time in the afternoon, check out Brunetti in the CBD for a cake & coffee/hot chocolate if you have a sweet tooth. Maybe Koko Black (CBD) or Ganache (South Yarra) if you really like chocolate.
Dinner being the hard part. I'd say head down to St Kilda, have a wander around the beach at sunset and find a cafe/restaurant around there to try out.