I forgot to put this in yesterday, but we had a press release ages ago promoting Historic Scotland's Doors Open weekend for St Andrew's Day. These places across Scotland will all let you in for free on Saturday and Sunday.
Edinburgh-specific, if you are busier than the sun and have no time to look at the list, are Craigmillar Castle and Edinburgh Castle. The latter is extending the event to St Andrew's Day itself, which is Monday 30th. That's three whole days of historical goodness. Are you excited? Of course you are.