Is the female student obligated to pay the driver the full monthly fare agreed upon, despite his absence for most of the month and his failure to provide the service completely, especially given that there was no prior agreement on how to calculate the absent days?

The aforementioned driver is considered a shared employee, and the rental agreement was for the service of transporting you to your workplace. This employee is not entitled to the wage merely by making himself available, but rather he is entitled to it in exchange for the work he has performed. Therefore, you are not obligated to pay the full amount; instead, you only pay for the days you actually attended work.