Although the Japanese "studless" snow tyres grip incredibly well, care should still be taken when driving on snow. In general you'll be fine if you simply keep to a low speed and don't perform any sudden manoeuvres. In particular brakes should be applied gently and definitely not while cornering.
There is a zero tolerance policy on drink-driving in Japan. Please call the Daiko service if you've had a few to drive your car home. Most restaurants and bars will be happy to call them for you, or their local number is 0261-72-6999.
All drivers are required to have either a Japanese License or an International Drivers License while driving in Japan. International Licenses are simply a translation of your country's license and can be obtained without a driving test in your home country. Please note you must bring your original Drivers License and International Drivers License when renting a car.