The number 27 goes from Hanover Street, the 8 from the Bridges or from down beside John Lewis (exiting the station via the back way under Waterloo Place gets you down Leith St without going up the Wavrley steps).
None of the buses stop at the West Gate, which is where the restaurant is. They stop in Inverleith Place on the far side, so there is a walk through the gardens. OK if it is not raining. The alternative is a 29, which is just as long a walk, but from a point a few streets away to the southwest - or a taxi to the West Gate.