Teaching you... business schedules
At first glance, it seems like a gift that most eikaiwa in Japan schedule your work days from Tuesday through Saturday, leaving Sunday and Monday off. Despite appearances, however, this is not done to give you a Monday available for normal business hours. Indeed, Monday is the Sunday of Japan, when many businesses are closed. National holidays are frequently on Mondays. A Monday-Friday workweek will give you more three day weekends and flexibility.
On January 21st, teams from all 47 prefectures of Japan (useful fact) will race over a 48 km course. I'm not exactly sure of the criteria for racing, but it's a safe bet you need to be a professional runner. Either way, this should be an interesting spectacle. Come to Hiroshima Peace Park at 12:30 for the start.
Official ekiden site
Miyajima Oyster Festival
It's a shame that the Sapporo Yuki Matsuri and this event are scheduled on the same weekend... I'll be sure to petition the country to ask they be separated next year. Come to the island of the famous floating torii and enjoy eating fresh oysters until you pass out, or take advantage of the aphrodisiac qualities and find a significant other for the evening. February 10th-11th.
It hasn't started snowing in Hiroshima yet, but the coldest time of the year should begin in February. Cherry blossom season starts in late March, early April. Almost every place in Japan is wonderful during the unofficial national flower season; remember, the best sights are one week after the blossoms start opening, in full bloom. Invest in a decent camera; these flowers don't last too long. My recommendations - the path leading from Saijo to Hiroshima University. Himeiji Castle. Kyoto, at the famous "rock-splitting" cherry blossom.