So, there are several things on which to comment here.

Ultimately, Hebrew and Arabic have a common ancestor both linguistically and palaeographically. Semitic languages based in alphabetic scripts (with the exception of Ugaritic) have been written R>L for a very, very long time. (Ugaritic, however, being a language represented in alphabet based cuneiform, antedated some serious changes in writing and orthography in the Semitic languages which I will discuss below.)

So, the first thing that we have to consider is the fact that we don't have any evidence for HEBREW as its own, distinct language until the 8th century BCE. By this point, however, writing conventions had been well established and scribal training and curriculum rather rigorous and fixed. Simply put, there was a standard of practice. Prior to the emergence of Hebrew as a distinct Northwest Semitic language, the epigraphic evidence we have really only attests to Phoenician (or, perhaps even more broadly, Canaanite).

Interestingly enough, proto-Canaanite was written vertically during the Late Bronze Age (ca. 15th-13th centuries BCE). We might readily consider the so-called Proto-Sinaitic inscriptions and, even, the inscriptions found at Wadi el-Hol. Turns out, the development of the script used to write Canaanite during this time period most likely derived from Hieroglyphs (many of which, of course, were written vertically). So, the change you talk about happened. It just happened much earlier than you might expect. As the writing shifted from columnar to sinistrograde, the graphemes rotated 90*. Some would rotate again, some resisted rotation.

The Arabic script only arose later after these writing conventions had essentially become frozen.

A few resources:

Gordon Hamilton, The Origins of the West Semitic Alphabet in Egyptian Scripts

W. F. Albright, The Proto-Sinaitic Scripts and their Decipherment

F. M. Cross, Leaves from an Epigrapher's Notebook