So this is a funny question, because that period is quite fuzzy and there's not very much hard evidence. Thanks for getting me to do some research, as it's certainly an interesting topic.
So you're talking about 1000 BCE southern Arabia, this would be the late bronze age and in what we now call Yemen was a mixture of many different peoples leading various lifeways. Along the coast there were bronze age kingdoms of village farmers, the largest being the kingdom of Saba which formed as early as 1200 BCE but at least by 800 BCE. In the interior you had (also bronze age) semi-nomadic pastoralists who had various animals but probably focused on cattle (and had horses).
This was quite a stressful time to be in nearby Mesopotamia, as this period (1200-800 BCE) is the change from the late Bronze Age to the early Iron Age. This is a fascinating and a disruptive time as the old order was re-made: nomads on the borders of Mesopotamian empires were invading and conquering local rulers, and the Egyptians pulled back from both of their colonies (Nubia and the Levant). Add to the mix that some groups had iron weapons and others were still using bronze, and you’re left with a very dynamic picture.
In Arabia people would’ve been more or less unaffected although most likely trade routes were disturbed. Goods originally flowed to what is now Djibouti and then onto Egyptian merchants trading in Punt (or overland to Ethiopians and Nubians). As the Egyptians had pulled out of Nubia and suffered extreme political crises it’s likely that far-reaching traders (such as those who would go so far as Punt) were cut back, interrupting regional trade. Similarly in the north the steady flow of goods from Mesopotamia was likely interrupted, and then aggravated by the rapidly changing political fortunes of northern Arabian border nomads. This also likely disrupted trade routes. All of this didn’t mean much to an average farmer or pastoralist in southern Arabia, although I would think (due to the work of Earle and Kristiansen) that changing trade routes meant changing centers of power.
While southern Arabia wasn’t directly effected by say, the collapse of a Mesopotamian empire, it was a part of the larger Near Eastern technological sphere. There were similar changes also happening in southern Arabia. During this time (ca. 1200-800 BCE) you see the switch from bronze use to iron. Also you see the first evidence of the domestication of the camel, first domesticated in southwest Arabia. Thirdly you see new irrigation technologies being employed which allow southern Arabians to live in what are termed city states. Due to increasing population density these feuding city states form larger polities, the largest of which was Saba. This was similarly quite a disruptive time in Arabia, with new metals, new animals, and new hierarchies forming.
But to get to your question: yes, Arabians in 1000 BCE either knew of horses or had domestic horses. Before we can say if they were ridden, we must first see how horses were introduced and used in Arabia. Domestic horses are first seen attached to chariots, and are first brought into the Arabian sphere by the Hyksos who ca. 1700 BCE invaded Egypt from the Levant. Perhaps horses and chariots spread through all of Arabia very rapidly starting sometime after 1800 BCE. There’s a possible analogous situation in the Americas historically. Horses were traded by the Spanish to Puebloans in the southwestern United States by 1600 CE. Within 170 years they’d spread from there to practically the rest of north America, being found in northern California and southern Canada. This transfer of horses + knowledge all happened without any Spanish handlers. Perhaps a similar rapid spread could’ve happened in Arabia, albeit with the pastoralists of the Arabian interior as the middle-men.
Also during the 2nd millennium BCE we see rock art depictions of the chariot throughout Arabia: all of this suggests that southern Arabian bronze age farmer chieftains became maryannu (the Mesopotamian term for chariot-centered nobility) let’s say sometime around the middle of the millennium. But of course the question you asked was “were people riding them” and that’s a tough one.
As horses were smaller and sometimes weaker in the bronze age it’s normally thought that they were only used in warfare hitched to chariots. But what this commonly touted fact means is only that they couldn’t support an armored soldier; presumably horses were ridden by normal people. One of the reasons they were domesticated in Kazakhstan by the Botai culture ca. 3500 BCE is because when ridden you can corral many more animals. In the late bronze age (13th century BCE) the Egyptians began to experiment with cavalry regiments (I presume being armored). So yes, I think people in southern Arabia in 1000 BCE rode their horses.
But perhaps you meant “did southern Arabians ride their horses into battle in 1000 BCE?” That is a very interesting and tricky question. Arabian rock art shows two main styles of warfare between 2000-1 BCE: chariots in the 2nd millennium BCE, and cavalry in the 1st millennium BCE. Presumably this switch in warfare was part of larger social change happening in southern Arabia in the late bronze age to the early iron age: where bronze using chariot riding maryannu were usurped by iron using horse-lords. Now importantly, we shouldn’t assume all images of one category happened during only that period, it’s very likely that some specific image of a chariot is from only 800 BCE while an image of a cavalryman is from 1500 BCE. So to answer this ancillary question, 1000 BCE is just at the split between when southern Arabians were switching from chariots to cavalry – so some groups did ride them into battle.
Sorry for such a long winded way to say yes.
Sources:
Horses first seen in Arabia after 1800 BCE (David Anthony) http://archive.aramcoworld.com/issue/201203/discovery.at.al-magar.htm
Horses in Arabian Rock Art - http://saudi-archaeology.com/subjects/horses/
Warfare in Arabian Rock Art - http://saudi-archaeology.com/subjects/warfare/
Arabian Bronze Age - http://arabian-archaeology.com/arch4bronze.htm
Arabian Iron Age - http://arabian-archaeology.com/arch3iron.htm
The anecdote about the spread of horses in north America - http://www.karimsadr.com/resources/Sadr%202008%20SAH.pdf
Rock Art of Saudi Arabia, by Majeed Khan www.mdpi.com/2076-0752/2/4/447/pdf