sleep, read the newspapers and 'play' with Akihito, if you get what I mean *wink*
There's a whole extra to tell you what he does with a day off:
http://www.mangago.me/read-manga/you_re_my_loveprize_in_viewfinder/mh/v000/c001.2/4/
There have been only two people who have ever called Asami by his first name Kuroda and Arbatov. why did Arbatov do that on the boat to annoy or is him?, because of family ties? Look who shows up.
Does Asami ever get any downtime for himself? Like on weekends and such? I wonder what he does to relax... Sleep all day long? lol