does the vertebral artery go all the way up to the anterior of the medulla oblongata near the pontine artery?

"The" pontine artery is an artery I am not familiar with. It is true that the vertebral arteries enter the skull and then join one another to form the basilar artery, which does lie upon (or better said, in front of, rostral of) the pons. Before they do, they each also give 4 seperate branches. 2 of those run forwardly, in front of the spinal cord, running back down; these are the 2 anterior spinal arteries. The 2 posterior ones fuse quite immediatly and form 1 posterior spinal artery, in the middle of the posterior groove of the spinal cord (they arise higher than the anterior ones, so closer to the basilar artery).

 

Now, the basilar artery itself is - to me - one of the most important arteries of your central nervous system. If it shuts down, it's bye bye (unless it's discovered and treated incredibly fast) - you may appreciate damage to the brain stem to be associated with depression of normal breathing, of heart rates and, eventually, of conscience (aka coma, and ultimately, death).

The vertebral arteries also give the bilateral posterior inferior cerebellar arteries (PICA).

The anterior inferior cerebellar artery (AICA) is derived from the basilar artery, as are the superior cerebellar artery (SCA) and the posterior cerebral artery, which then communicate with the arteries taking care of more rostrally found brain structures.

 

But now, the ultimate answer to your question: the "rami ad pontem" (branches at the bridge/pons) are the branches delivering blood to the pons. They arise from the basilar artery, and, by extension, from the vertebral arteries.

 

Ta-ta!