In general, Japan is a safe country and the crime rate is low, but you still need to be careful about theft, robbery, pickpockets, etc. If you have lost something or become lost yourself, please go to the nearest Koban (police box) for help. Japanese policemen are typically very kind and will show you a map, or help you by going through their lost and found list. But please note they do not speak very good English usually, and they may also ask you to show them your residence card or passport.
You do need to be careful of bicycle theft, and if you are riding a bicycle (especially at night without a bicycle light on), a policeman may stop you and ask some questions. It is a requirement in Japan that bicycles are registered, and a policeman may check if the bicycle is registered under your name.
Tap water is normally safe to drink in Japan and water fountains are available in many places (stations, parks, public facilities etc.). Bottled water can also be easily found in vending machines and convenience stores etc, at reasonable prices.