Depending where you're going, a car is probably your best bet.
By bus
Night buses go from Tokyo to Yurihonjo. Several bus companies offer this service. Check the websites or ask at the bus station for more information.
By train
From Ugohonjo Station in Honjo, the Akebono night train goes direct to Tokyo, and the Nihonkai night train goes direct to Kyoto. The nearest shinkansen stations are the Komachi Shinkansen in Akita or Omagari, and the Tsubasa Shinkansen in Shinjo, northern Yamagata. Both of these shinkansen lines go to Tokyo. There are many daily trains going north and south out of Honjo. There is also a private train line that goes from Ugohonjo Station to Yashima, by way of Yuri. See hyperdia for train schedules.
By car
Within Akita, cars are the most convenient way to get around. Highway 7 connects Honjo with Akita to the north, by way of Iwaki, and Nikaho to the south, by way of Nishime. Highway 105 connects Honjo to Daisen in the NE, by way of Ouchi. Highway 107 connects Honjo to Ugo and Yokote in the east, by way of Higashi Yuri. Highway 108 connects Honjo to Yuzawa in the SE, by way of Yuri, Yashima and Chokai. There are several small highways through the mountains, but some of them are seasonal or are closed for long-term repairs, so ask someone locally and check. In the winter, the roads can be snowy and icy, and the mountain areas — Chokai, for example — get tons of snow.