Was Gautama Buddha a believer of the religious beliefs prevalent in his time and in his cultural context? Before Siddhartha Gautama became The Omniscient Buddha, did he believe in, support and uphold the religious beliefs that the commoners and/or learned persons practiced at the time that he lived?

Or alternatively, did he question the established religious beliefs of his time, and in his culture?

In other words, did Siddhartha Gautama arrive at his Enlightenment by BELIEVING in prevailing religious mythology and principles, or did he arrive at his Enlightenment by QUESTIONING, EXAMINING and REJECTING them?
There's no belief for a Buddha. A Buddha is omniscient. He knows what is right and what is wrong by direct knowledge.

Yes, the Buddha did uphold certain religious practices of the time which are congenial to the path.

ex: The five precepts, 13 Dutangas, attainment of the Janas

He also rejected many false views which the contemporary religions were based on. They are grouped into 62 in Brahmajāla sutta.