All perfect praise be to Allaah, the Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
If you were observing a voluntary fast that day on which you broke fasting, then you would not be required to make up for that day according to the majority of scholars although some scholars are of the view that making up for that day is nevertheless obligatory. The first opinion, however, is the correct opinion because it was narrated by Muslim that the Prophet, sallallaahu ‘alayhi wa sallam, once intended voluntary fasting and then broke it. It was not narrated that he made up for that day. There are also many other proofs in this regard stated by the scholars.
If you were observing obligatory fasting, such as in Ramadan, or making up for a missed day, or fasting for a vow or expiation in which the sequence of fasting is not a condition, or is a condition unless it is interrupted for a valid excuse (according to the preponderant opinion), or that you think that it is obligatory to make up for the voluntary fasting which on breaks, or you follow the opinion of whoever thinks so among the scholars, then you are required to make up for that day.
As for your double intention of making up for the day you missed and fasting the day of ‘Arafah, then the preponderant opinion is that this is approved because the Sharee‘ah objective of performing that act of worship was realized. The purpose was to fast the day of ‘Arafah and this indeed took place.
This is similar to when one performs Ghusl (ritual bathing) on Friday for Janaabah (post sexual intercourse impurity). This would be enough for him both for the Janaabah and Friday Ghusl, according to the four Imaams of Fiqh (jurisprudence).
The prominent scholar, Ibn ‘Uthaymeen said, "The person who fasts the day of ‘Arafah or the day of ‘Aashoora’ (the 10th day of Muharram) while he is still required to make up for missed days in Ramadan, his (voluntary) fasting is valid. If he has a double intention of fasting on that day voluntarily as well as making up for the missed day of fasting, he will get the two rewards, the reward of fasting the day of ‘Arafah or ‘Aashoora’ along with the reward of making up for the missed day of fasting. This applies to voluntary fasting which is not related to Ramadan."
Nevertheless, it will be better if you make up for the missed day of fasting to avoid doubt and to be on the safe side.