Will I be sinful, and is it obligatory for me to fast, after I swore to my sister—while I was angry—that I would refrain from giving her what she asks for and fast three days for her sake if she did not obey me, then I gave her what she asked for later?

If you swore by Allah or by an attribute of His, and then broke your oath due to your sister refusing to give you what you wanted, then you must offer expiation for breaking an oath. The oath ended when the breaking occurred, and there is no sin upon you after performing the expiation. The expiation is feeding ten poor persons or clothing them, or emancipating a slave. And whoever cannot find [the means] - then a fast of three days, as stated in verse 89 of Surat Al-Ma'idah. Breaking the oath with expiation is better than fulfilling the oath and severing ties with your sister.