Was Gautama Buddha a believer of the religious beliefs prevalent in his time and in his cultural context?
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.