National Gallery. If the Three graces are there, then that is worth seeing on its own. A quick google doesn't show if it is there or at the V&A at the moment.
Look at a small bronze statue of a man on horse, then go to the Royal Mile, opposite the heart of Midlothian, and see the full size statue.
Greyfriars Kirk to see the tombs and mausoleums, possibly a visit on the Sunday for a service in Gaelic?
The National Museum. If you have read Rebus, you'll know of the tiny dolls found near Arthurs Seat, which inspried him to write about them. Go up to the top, open garden for great views.
A walk up to Arthurs Seat. Well, ok, get a Taxi up, then walk down.
Calton Hill and take a 'classic' photo looking down Princes St.
Walk up the 200+ steps of the Scott Monument for great views too. Wonder at the fat Americans who think they can get up.
Have a picnic in Princes St. gardens. Look up local info to see if a band are playing on the stand.
Wander round the Old Town, looking in the tiny closes. Mary Kings Close is the best, blocked off for 150 or so years, now open, but book in advance only iirc.
Have a pint in one of the many bars in the Old Town.
Avoid the Royal Mile whenever possible, unless you want to buy some tartan branded tat.