What is the ruling on someone who vowed to recite Surat Ya-Sin daily, then forgot it on one of the days and only recited part of it? And how can he be absolved from this vow?

Whoever vows an act of obedience to Allah must fulfill their vow. Reading only a part of what was vowed to be read in full is not sufficient. They are obligated to make up for what they missed. There is no way to be absolved from this vow by offering expiation as long as one is capable of fulfilling it. While such a vow is initially disliked, it becomes obligatory to fulfill once made.