Practical information : Sightseeing Japan
Times vary greatly, but most museums and parks will be open between 10am and 5pm. On Saturdays and Sundays, there are often long lines, especially for temporary exhibitions. Each institution has a variable closing day. Temples and shrines open early (around 8am) and also close at 4pm or 5pm, unless there is a festival.
Smoking is prohibited in the vast majority of public places.
In museums, small shops, workshops, it is fashionable to do things yourself (origami, paper, wooden objects, etc.). If some of them offer a real opportunity to try Japanese handicrafts, others are sometimes way too expensive for what is offered. You be the judge.