I can think of two basic answers: one historical, the other theological.

The historical answer concerns the postwar ascendancy of the Liberal Democratic Party (LDP), which has held power almost continuously since 1955, including the prime ministry, with only a few brief interregnums of opposition party control. Although there's nothing about the LDP's center-right, market-based politics that inherently align it with Christianity, its long entanglement with, and clandestine financing by, the United States is another matter. I've talked about this in some detail in another thread. But the point being: especially in the early postwar decades when US influence on the LDP was the strongest, Christian Japanese gained a kind of leg-up for their presumed anti-Communist allegiances.

The theological issue is simpler. Christianity, unlike the widespread forms of Buddhism and Shintoism practiced* in Japan, is an actively evangelical faith. That is, it pushes its adherents to become directly engaged in their communities and concern themselves with the well-being (spiritual, of course, but also, often, physical/material) of their fellow citizens. This civic-engagement and community-mindedness is, fundamentally, what politics is about. It's only a small step from there to electoral politics per se.

For more on Japanese Christians and civic engagement, check out Sheldon Garon's Molding Japanese Minds. John Dower's Embracing Defeat gives a good account of the postwar formation of the LDP.

*It's debatable the extent to which Buddhism and Shintoism are "practiced" in the same way that Christianity is, at least to lay believers in Japan. But that's an issue for another day!