How authentic is the Hadeeth that states that fasting the Day of ‘Arafah atones for the sins of two years and fasting the tenth of Muharram atones for the sins of one year?
All perfect praise be to Allah, the Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His Slave and Messenger.
This Hadeeth is authentic and it was narrated by Muslim on the authority of Abu Qataadah, may Allah be pleased with him. The Hadeeth mentions that the Prophet was asked about observing fast on the Day of ‘Arafah and he said: "It atones for the sins of the preceding year and the coming year." Morevoer, he was asked about fasting the tenth of Muharram and he said: "It atones for the sins of the preceding year." This Hadeeth was also narrated by Ahmad and Abu Daawood on the authority of Abu Qataadah, may Allah be pleased with him.
Thus, the Hadeeth that you mentioned is authentic and it is a Marfoo‘ Hadeeth (i.e., directly attributed to the Prophet ), but the particular wording that you mentioned is not among the authentic wordings of the Hadeeth despite the fact that their meaning is correct.
Allah Knows best.