Dear Rudi. I can offer free internet from my place. Walkable from Downtown and even from East Melbourne.
For public access look at
www.yellowpages.com.au Search for internet cafes.
I have a list in my street directory too:
Hawthorn:
Cybernet Cafe
789 Glenferrie Rd, tel.9818 1288
City Centre:
Go Local Internet Lounge
277 Little Lonnsdale St, tel.9639 6060
Mary Martin Bookshop Cafe
108 Bourke St, tel.9663 9633
Melbourne Central Internet
Level 2, Melbourne Central off Swanston Walk, tel.9922 1100
Myer RMIT Internet Cafe
295 Lonsdale St, tel.9661 1700