I know you're going the other way but from the airport there's nothing until Haymarket (there might be a restaurant type place with a bar at Edinburgh Park Central but I doubt its what you're looking for).
Haymarket has 3 decentish boozers, Platform 5, Ryries and the Haymarket, very close by.
West End they're not so close but there's a couple in William Street and in Shandwick Place.
Princes Street you'll have to go up into Rose Street or George Street.
St Andrews Square you have the Abbotsford (in Rose Street) or the Cafe Royal and the Guildford. Many other options are available.
Picardy Place there's a couple opposite the platform. Mathers at the top of Broughton Street or the Theatre Royal are my recommendations. There's a few LGBTQ bars there too, I've not been in any recently to say how decent they are.
MacDonald Road has the Windsor (my local so it must be fabby), the Joker and the Thief and what used to be Vitoria is now a bistro bar with lots of outside seating. There's a few others closeish.
Balfour Street has the Harp and Castle, Robbies. Loads of others close by.
Foot of the Walk. If you hate Wetherspoons avoid the pub called the Foot of the Walk! I'd also avoid the Marksman too. There's a few others along Duke Street (nothing special) or you could experience the Central
There is a pub at the North end of the platform but I can't remember it's name, it used to be OK.
The Shore isn't on the Shore its a few minutes walk but there's plenty of pubs by the platform including the famous Port o Leith and much better places to eat than OT. Round the corner to Marmalade Cafe for a superb all day breakfast for example
Port of Leith isn't really the Port o Leith and has no bars nearby.
Ocean Terminal doesn't have any bars nearby.
Newhaven isn't really in Newhaven. If you walk along to Newhaven village there's a couple of pubs.
Good luck.
