Growing up, I had learned that Eleanor Roosevelt influenced President Roosevelt to protect Kyoto and Nara from bombing. I know one other person who had heard the same thing. Now, however, while I can find information about the influence of Langdon Warner and Henry Stimson, I can't find anything about Eleanor's influence on the decision. Also, I had heard that there was a secret agreement between the U.S. and Japan that protected these locations from bombing so long as they weren't used for military purposes, but I can't find anything about that either. Are these just myths?
The non-bombing of Kyoto has for whatever reason accreted many myths and rumors over the years, frequently by the Japanese themselves. The archival record is very clear though: it was Henry Stimson (and essentially Stimson alone) that got Kyoto put onto a "do not bomb" list. Which is to say, not Eleanor Roosevelt, not Langdon Warner, and definitely not some kind of US-Japanese pact. The decision not to bomb Kyoto came during the Truman administration, not Roosevelt's.
It is worth noting that Kyoto was used for military purposes during the war. There are complicated reasons why Stimson wanted it preserved from bombing, but ultimately they were probably personal. My own reading is that Stimson (for personal reasons) saw the preservation of Kyoto as the one small mercy he could grant Japan; he was not a fan of the indiscriminate bombing campaigns but lacked the power to stop them.
I'm not sure why there are so many rumors about this, especially since the archival record is not particularly difficult to trace anymore. The Japanese in particular have really built Warner up into a hero in their mind, despite Warner's own protestations. (I gave a talk on this in Kyoto a few years ago and they were still really shocked; no American historian of this thinks Warner had anything to do with it — that myth was put to rest 40 years ago, if not earlier.)
I don't know of any order not to bomb Nara. The "reserved areas" list was very short: Hiroshima, Kyoto, Niigata, Kokura. That's it. (Not even Nagasaki!) And as you can probably see, Kyoto was the only one on it that was there because it really wasn't to be bombed — the others were atomic bombing targets, as originally was Kyoto until Stimson intervened. Some sources say that Nara was firebombed during World War II, with some 66% destroyed, but I don't have ready confirmation of that.
It is perhaps a bit ironic — or understandable — that the figures the Japanese have latched on to as possible saviors all include people of high culture and not someone like Stimson (a soldier/bureaucrat). But such is the weirdness of history.