if I was you.......i'd ask the person you are picking up to get on a train at waverly and head out to Edinburgh park or the gyle. They are very regular and will save you about 20 minutes getting into the town and another 20 getting out.
failing that.......get the person you are meeting to meet you outside the station on market street (just outside the Tattoo office and next to the Edinburgh Art centre)
if you take the M8 to hermiston gate follow directions for city centre - A71, that road is a dual carriageway, go straight over teh 3 roundabouts, follow that road and you'll pass an HSS hire shop on your right, move into the left hand lane then
at the traffic lights, go left and at the roundabout go right (2nd exit) follow that road all the way - it will take you to the western approach road which you can follow all the way until it ends on a t junction on Lothian road.
turn right and then 1st left at the next lights and follow that road down to the grassmarket, through the cowgate and then turn left after the travel lodge. Go up the hill, straight across the royal mile and follow the road round to the left - and you are on market street
seriously, get your pal to jump on a train to Edinburgh park!!