What should someone do who takes medication for sinusitis that causes drowsiness and makes him miss prayers?

You must be determined to perform the prayer on time and take appropriate measures to wake up. If sleep overwhelms you, there is no sin, for the Prophet, peace and blessings be upon him, said: "There is no negligence in sleep; rather, negligence is in wakefulness, that one delays a prayer until the time for another prayer enters." It is not permissible to sleep after the time for prayer has begun if you believe you will not wake up until after its time has expired. In all cases, if the time expires before you wake up, you are obligated to make up the prayer, for the Prophet, peace and blessings be upon him, said: "If any of you oversleeps or forgets a prayer, he should pray it when he remembers it, for Allah says: 'Establish prayer for My remembrance.'"