Probably the oldest example — being one of the oldest known examples of writing to begin with — is Sumerian cuneiform writing.
Like Chinese, Sumerian cuneiform was originally written in vertical columns from top down and right to left, but sometime around 2000 BCE the writing rotated 90 degrees counterclockwise and began to be written in horizontal lines from left to right and top down, just like we write English today.
Unlike with modern Chinese, we can't blame this shift on the influence of the Latin script, since that wasn't invented for a few more millennia yet. The only other writing systems in widespread use anywhere nearby were Egyptian hieroglyphs and the related hieratic script, which I guess might have been an influence, since contact between the Sumerian and Egyptian civilizations certainly did exist despite the distance and the inconveniently placed desert in between. However, that's just wild speculation; no definitive evidence on the cause or causes of the cuneiform writing direction change, and whether it was a local innovation or influenced by other writing systems, is currently known.
Left: Cuneiform inscription written in vertical columns on a statue of Gudea, ruler of Lagash, c. 2120 BCE. Detail from photo on Wikimedia Commons taken and released into the public domain by Marie-Lan Nguyen. Right: (Mostly) horizontal cuneiform inscription on a statue of king Idrimi, c. 1600 BCE. Detail from photo by author.
Also unlike Chinese, when the direction of cuneiform writing chanced, the signs rotated with it. A peculiar consequence of this is that the change is unobservable in texts written on clay tablets (which comprise the majority of surviving cuneiform texts) since they could be held equally well in either orientation*. The only place where it can be seen is in inscriptions carved on walls and statues and other objects with a definite up–down orientation. This also makes dating the change more difficult, not only due to the limited evidence, but also due to the fact that monumental inscriptions were often written in a deliberately archaic style, and quite likely preserved the old vertical writing order long after it had otherwise fallen out of favor.
Also, the change was obviously gradual, and both writing directions coexisted for a long time. Indeed the vertical writing style never quite entirely died out (until cuneiform writing as a whole did), especially in archaizing contexts or where vertical text just happened to fit better. Sometimes both could even be found together, as in this illustrated stone tablet (BM 90922) from c. 875–850 BCE, where the main body of text below the image is written horizontally (as usual by then), but the names of the figures above are written vertically next to each of them.
Ps. Having practiced cuneiform writing a bit myself, one relevant observation I've made is that, at least for me, the most natural and convenient way to hold a tablet while writing on it is actually at a roughly 30° to 60° angle between horizontal and vertical, so that one ends up writing diagonally from top left to down and right(!). Also, especially when writing on a small hand-held tablet with an old Sumerian style narrow stylus, where the same edge of the stylus is used for both vertical and horizontal wedges, it's often convenient to rotate not just the stylus but also the tablet when switching between wedge directions (which one generally needs to do at least once per sign, if not more). So, regardless of the orientation in which the tablets were read, I'm pretty sure that the actual direction in which they were written could vary considerably even during the writing process.
*) Clay tablets could have seal impressions with pictures on them, but the seals were not always applied in a consistent orientation either, limiting their value as evidence of the intended reading direction. Actual drawings on clay tablets are relatively uncommon, and most that I know of tend to be schematic in nature, e.g. maps or geometric figures that also have no obvious intrinsic orientation.