The government headed by the prime minister is mainly in charge of everything, especially the day to day matters, Assad's main focus is foreign policy, the army, national security, in addition to overseeing the conduct of the minsters, their aides and deputies and their head, the prime minister, naturally.
There's a minister of defense under the Prime Minister in the cabinet and a chief of staff and Assad is the commander in chief.
There's a minister of foreign affairs under the PM in the cabinet, all the diplomatic body under him and Assad is the top diplomat of the country.
There's a minister of economy and trade under the PM in the cabinet with all the tasks entitled to his post and Assad is the top decision maker in cases that need his intervention since he oversees other sectors not under the control of the minister, which the PM is supposed to take care of otherwise.
There's a governor for each of the 14 provinces making up Syria, they report to Assad directly but at the same time the PM oversees their work through the departments of each of the ministries in each of the provinces, like the department of health in Aleppo under the ministry of health under the PM where the governor oversees their work and is in direct charge of its work, these departments report to their ministries under the PM and at the same time the governors oversee their work and report to Assad periodically.
And so on.
We have a presidential system in Syria and you can consider the prime minister as the second top executive office in the country, the prime minister swears to abide by and protect the constitution in front of Assad not in front of the Parliament, Assad swears the same in front of the parliament. In other words the cabinet are appointed by Assad to do the designated work within their domains, the PM should take care of all aspects of all these ministries and report to Assasd when needed, the president also heads the ministers meetings when he's present where the PM acts as his deputy in his presence, and when not in his presence the PM is the top guy in those meetings.
I hope I didn't complicate it much, just say the executive office is under the PM and the overall office is under the President.