Many cafes and bars offer free wireless. If you're around the east end or in Leith try Joseph Pearce's or one of his sister pubs further down the walk. No real ale unfortunately but coffee and continental lager with a slice of kitsch.
the library on George 4th bridge at the top of the mound has free internet.
Monbodo on Bread Street has free wifi for the price of a cuppa/drink.
Black Medicine coffee company on Marchmont road and nicolson street has free wifi, as do a lot of the student coffee places around the Uni in Bristo Sq.
I sell a handy illustrated guide to Edinburgh on my website http://www.clairemurray.co.uk