Except visiting Temple and Shrine, another New Year tradition in Japan is watch the first sunrise of the year! Many people prefer to do this beside sea and it is absolutely possible at Amanohashidate area. (Picture is from 1st Jan 2013 but you can guess I was too lazy to see the sunrise ;) )
Usually, you can take a cable car to go up to Kasamatsu park to see the sunrise. The cable car is run from 6.30 in the morning, only special for the New Year's day!!
Kasamatsu Park will be open on 1st January 2014 from 6:30AM until 16:30.