Is the hadith, "Whoever fasts the last day of Dhul-Qa'dah and the first day of Muharram has concluded the year with obedience and commenced it with obedience, and Allah Almighty will forgive him the sins of 50 years," authentic?

No hadith with this meaning has been reported, and it appears to have no basis in the recorded books of Sunnah. The aforementioned book is not among the authoritative books of Sunnah. If a hadith contradicts reason or established narrations, or if it goes against fundamental principles, then it is fabricated (mawdu‘). The meaning of contradicting fundamental principles is that it falls outside the main collections of Islamic traditions, such as the Musnads and well-known books.