Is zakat due on the sum of (300) thousand Riyals, or on the (100) thousand Riyals remaining in the bank after a year has passed, or is zakat not due on the entire sum due to the absence of a monthly income?

Zakat is obligatory on money that has been in your possession for a full Hijri year and has reached the nisab, whether you are investing it or saving it for expenses. If a year passes and some of it remains with you, then you must pay Zakat on that remaining portion. There is no Zakat on what you spent during the year. This applies if the money was a gift from the employer.