Have a pint with me :lol: Forbidden Planet on South Bridge is probably the best bet for manga (everything on 3 for 2, not just Viz stuff). They don't have any anime though. CeX and Fopp are almost right next to each other on Rose Street, and HMV and Waterstones are at the West End of Princes Street. There's also a shop called Anime Republic in Princes Mall by the station. They sell merchandise, and a bit of manga. They have a lot of bootleg stuff but some good stuff too.
As for free/cheap places to visit, it's not terribly expensive to climb the Scott Monument, but you do need a head for heights and not mind small dark places! You can walk all the way up to the castle, but it is about £15 to get in. Just walk up for the view and then come back down. If it's Sunny, the Camera Obscura is great fun (also on the Royal Mile)
I will try and think of more later!