What is incumbent upon me if I swore to my brother that I would annoy him during his computer game if he did not do a certain thing, and then I did not fulfill my oath? And do I owe expiation?

The question is not entirely clear, but it seems you swore to bother your brother if he played, and you did not. To fulfill the oath, it is sufficient that you do what you swore to do, and expiation is not required merely for the delay, unless there was an intention for immediate action.

However, if bothering the child is unlawful for you because you do not have the authority to discipline him, then you must offer expiation for your oath and refrain from bothering him, based on the saying of the Prophet (peace be upon him): "Whoever swears an oath and then sees something better than it, let him offer expiation for his oath and do that which is better." So, expiation is for an oath that is broken for a good reason, and also for an oath that involves a sin, where breaking it is an act of obedience.