Crucifixion was a common method of execution in Japan at the time, and was pretty much the default for serious crimes, especially those that had a whiff of treason about them, such as servants killing, wounding or falsely accusing masters, wives killing or wounding husbands, and children killing, wounding or falsely accusing parents. Adhering to a prohibited religion would have been seen as treason, and crucifixion would have been an expected method of execution.

The bar for a crime to be serious enough for execution by crucifixion was quite low - forging coins, sneaking past roadblocks, and letting foundlings die were also punished by crucifixion. "Mild" capital punishment (simple decapitation) was reserved for relatively minor crimes, such as theft, fraud, adultery by a wife, forgery of documents, unsuccessfully attempting to wound a master, etc.

There were other forms of severe execution, such as sawing (which usually involved burying the prisoner in the ground up to their neck, and then sawing their head off), burning, boiling, and being torn apart by oxen. These were sometimes combined with crucifixion - killing a master was often punished by the combination of sawing and crucifixion.

According to Schmidt (2002), water-crucifixion, consisting of crucifixion below the high tide line, such that the water would reach the neck, was mostly applied to Christians. This form of execution was clearly limited geographically, and Christians formed a large fraction of the executed during the peak of their persecution, this might simply be a coincidence rather than water-crucifixion being aimed specifically at Christians.

Christians were also executed by sawing and boiling.

References:

Petra Schmidt, Capital Punishment in Japan, Brill, 2002.

NSFW: Photograph of Japanese crucifixion by Felice Beato, 1860s: http://www.getty.edu/art/collection/objects/241307/felice-beato-japanese-criminals-crucified-and-beheaded-english-1863-1868/