As a private American I would take his lead, him being the Emperor. I would bow if necessary, not that low but certainly watching him to go a slight notch lower, and I certainly would not extend my hand while bowing. It is all academic because I doubt he would bow to me and would probably do the Western style. He
might bow if I were sandwhiched in a group of two or three Japanese, in which case I would spin on a dime and do what those were doing around me, in perfect Japanese style and timing. But if one on one, it is also certain 100% it would be a mutual handshake. I would use my best
keigo (respectful form of Japanese language, but I would also not blab on and maybe say one short sentence and wait for him to ask ME the questions).I might, (in the handshake and me as a private citizen, not President representing an entire nation), naturally considering the level of such a person as Emperor, bring my head down a bit out of some respect but never a 90-degree at the waist, hand outstretched like that funny looking Obama individual. That has to be the most ridiculous-looking thing.
Now as President? I do what nearly all world leaders do and have done and should be trained in. Respectful, friendly handshake, no bow, eye contact maintained, warm, relaxed smile (everything Obama was NOT).