Have a drink in The Oxford Bar, 8 Young Street. It is the pub that the fictional Inspector Rebus drinks in. It's also the pub that his creator Iain Rankin often drinks in. Yes its just a pub, that is it's charm
Before you leave, visit Valvona and Crolla at the top of Leith Walk and stock up on food from an amazing Italian Deli (and it has a cafe too which is also great). When I was working at the fringe, around the time Trainspotting was set, I was living further down Leith Walk which was smackhead central (the book was right) that Deli was an absolute oasis in a sea of effluent picked up my lonch from there every day. It's been gentrified now but that place has been there for decades and it has the most amazing smell, go there just for the smell. I will never go to Edinburgh and not visit there before I go home