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I'm not aware of any long stay parking at the cruise terminal. there is a large multi storey facility at the Ocean terminal next to it but I don't know if they do long stay facilities or not. You could try contacting the Forth Ports Authority who own and manage all the Leith Dock area.
Waverley Station (Edinburgh's main station) is only about 2 miles from leith Docks and normally would be a fairly quick and cheap taxi ride (so no need ofr a transfer). However the centre of Edinburgh and leith Walk (main Road from Waverley to Leith) are in chaos at the moment (and wil be for months and months to come) with all the works on the roads for the new tramways that are being introduced in the next few years. There are other routes a taxi can take but generally speaking traffic is still heavy 9especially at rush hour)because of the knock on effect of road closures/restrictions on the other main roads. So I'd say allow plenty of time to get there. If you arrive too early, you could always "kill time" by doing the tour of the Royal Yacht Britannia and Visitor centre or a quick trip around the Ocean terminal Shopping centre 9quite new but not that big and all the usual chains). there are also some lovely cafes, bars and restaurants along the waterfront close to the Cruise terminal. Britannia Spice is the closest and has a great reputation (mix of all types of Asian foods) or alternatively half mile away is the very long established Waterfront Wine bar and restaurant or Skippers Bistro (both seafood restauarant although you can just go for a drink)Both have an amazing atmosphere and reputation.
Whichever way you decide to get to Leith, hope you enjoy yopur visit to our Capital City and have a fantastic cruise.
I am a disabled driver trying to book in my car for a week whilst I go on a cruise with fred olsen lines, I am having great difficulty in finding out where I can leave my car at the docks for a week, unfortunately fred olsen group not much help and I would not book with them again,very dissapointed thinking of cancelling the cruise.