Public Transport Directions
From the airport: (the information provided below is from the Edinburgh Airport Website)
Buses into Edinburgh
This express bus service runs from the airport to Waverley Bridge (near Princes Street and the main rail and bus stations).
Buy tickets at the airport information desk, at the bus stop, from the driver or in advance online
Frequency: Every 10 minutes (early services every 20 minutes).
Journey time: 25 minutes (Waverley Bridge).
Fare: Single £3.50, return £6 (reductions for children).
Visit the Airlink 100 website: http://www.flybybus.com/
Night bus N22
This service runs from the airport via South Gyle to the city centre and on to Ocean Terminal in Leith between 00:47 and 04:13, seven nights a week.
Frequency: Every 30 minutes.
Journey time: 28 minutes (Waverley Steps).
Fare: £3 flat fare (unlimited travel for one night).
Visit the Lothian night buses website: http://www.nightbuses.com/
This Lothian Buses service runs to the city centre and on to Ocean Terminal, with stops including RBS World Headquarters, the Scottish Parliament and the Palace of Holyroodhouse.
Buses run every 15 minutes Monday to Saturday daytime and every 30 minutes evenings and Sundays.
Frequency: Every 15 minutes.
Journey time: About an hour depending on traffic (Holyrood).
Fare: Single £1.50, day ticket £3.50 (reductions for children).
Visit the Lothian Buses website: http://lothianbuses.com/timetables-tickets-maps/timetables/35
I recommend that you get on the airlink 100 which will take you to waverley train station. From there, either take a taxi to western harbour breakwater or you can catch the number 10 bus which will take you straight to western harbour. The bus stop is at he St James shopping centre, opposite the OMNI centre, on leith street. It's right past Starbucks.
you could simply take a taxi from the airport to the flat, which would cost around £25.
You also have the possibility to hire a car from the airport.