The interesting bits are pretty compact, although quite hilly. The city centre lends itself exceptionally well to a slow mooch admiring the architecture, frequent café stops and browsing in interesting shops so you should be able to do plenty of that without feeling you've missed out.
It's worth getting down to Princes Street to see the gardens and look back at the old town and castle. Two free art galleries on the Mound (in the middle of Princes Street and on the way to/from the old town) if you're into that kind of thing.
The Castle itself is quite expensive, don't feel obliged, it's not the greatest castle once you're inside it. If you can make it up as far as the drawbridge you'll get good views.
Recommend from there mooching down the Royal Mile. You can get free tours of the Scottish Parliament (I think not Sundays, maybe worth checking).
The Museum of Edinburgh is free and although a bit dry it has a good audio visual of the history of Edinburgh.
You might swerve off, especially to go down Victoria Street to the Grassmarket (has an actual market on Saturdays) and up Candlemaker Row to see Greyfriars Kirk and Greyfriars Bobby.
Greyfriars Bobby is opposite Chambers Street with its massive free museum.
There are a few fairly reasonable museum type things along the way like John Knox's House, but I wouldn't get FOMO about it. Wandering the streets and popping into cafes is a good use of your time in Edinburgh.
The weather can change quickly so layers are recommended. Comfy shoes also recommended.