What is incumbent upon me so that I do not sin, given that I made a vow years ago to gift my teacher upon my success in a specific subject, knowing that I do not know his whereabouts now, and I was unaware of the methodology of vows at that time?

If the gift was intended as a bribe, then it is not permissible to fulfill the vow, and an expiation for an oath (kaffarat yamin) becomes obligatory, conditional on success. However, if it was intended to foster affection, then the vow must be fulfilled upon success. If the teacher cannot be found, then an expiation for an oath becomes obligatory.

As for abandoning prayer and fasting: prayer is an obligatory pillar and its performance is necessary [or, its making up is necessary]. Abandoning fasting is a grave sin and it is obligatory to make up for all past years.