I was there last month. Attractions open now might have to close in future, but assuming they're open:
Edinburgh Castle: tickets only available in advance and limited numbers throughout the day. Most of the inside parts were open last month. Portaloos instead of inside toilets though.
National Museum of Scotland: free but have to book in advance. Again limited numbers. A few galleries are closed because they can't maintain social distancing inside them, but most are open and a one-way system is in operation, which actually makes for a better visit.
Calton Hill (end of Princes Street): good walk and great views. Also the burial ground on the opposite side of the road.
Walking in general: through Princes St Gardens, St Cuthbert's graveyard, along Kings Stables Road and around the base of the castle rock, up to Johnstone Terrace and then the castle esplanade. Down the Royal Mile and explore the closes and courts.
Dean Village isn't far to walk and is very picturesque. Unfortunately the river path along the Water of Leith is closed in a lot of places, otherwise that would be a good walk.
All the above can be done without using public transport (assuming you stay in central Edinburgh).
I didn't make it to Arthur's Seat/Salisbury Crags.
And shopping of course!