Will a person who masturbated while fasting, knowing the impermissibility of his action, then completed his day refraining from food and drink with the intention of repentance, be rewarded? And is that considered half a day's fast?

If the fast was for Ramadan, then one must abstain for the rest of the day out of respect for its sanctity. However, if it was a voluntary fast, then it has been broken, and it is neither obligatory nor beneficial to abstain, because it is no longer a valid fast according to Sharia. You should hasten to perform a good deed that expiates the bad one you committed, such as performing ghusl (ritual bathing) and praying two rak'ahs with the intention of repentance, or reciting the Quran, or giving charity. You must repent to Allah for this vile act, strive against your lower self, and do not despair of Allah's mercy, for Allah accepts repentance no matter how great the sins. Allah Almighty says: "Say, 'O My servants who have transgressed against themselves [by sinning], do not despair of the mercy of Allah. Indeed, Allah forgives all sins. Indeed, it is He who is the Forgiving, the Merciful.'" And the Messenger of Allah, peace and blessings be upon him, said: "O son of Adam, as long as you invoke Me and hope in Me, I will forgive you for what you have done, and I do not mind." You should also adhere to dhikr (remembrance of Allah) and du'a (supplication), accompany good people, occupy yourself with beneficial deeds, and stay away from distractions.