Dip In Café
It could be hard to find a place with wifi on Bourke street because it’s full of touristic bars and restaurants, but if you walk a bit down this particular alley, you’ll find this gem in the middle of the city. The food can be described as superb and at the price is very reasonable for what you’re served. The atmosphere of the place is also very nice and the tables and seats are comfortable.
Unfortunately the biggest problem is the wifi connection, which is quite poor. Just ask for the password from the kind staff but don’t expect much in terms of performance. Many people that come here most likely don’t come for the wifi connection but rather the food and coffee instead.