"The important thing is, one day, night will become day for several weeks on Earth."

According to Wikipedia[1] the expected visible magnitude of the supernova is around -12mag. The full moon's apparent magnitude is around -12.74mag[2]. So we'd more likely have "another moon" for some time instead of a second sun.

[1] http://en.wikipedia.org/wiki/Betelgeuse [2] http://en.wikipedia.org/wiki/Apparent_magnitude