This question got buried a bit, but I saw it and wanted to respond.

Arabic would have had a relatively limited but prominent usage in Central Asia, both before the Russian conquest, and also during the period of tsarist rule to 1917, which was direct in some areas of Central Asia, but also indirect (ie, through nominally independent protectorates) in Bukhara and Khiva. Bukhara in fact was a center of Islamic learning that drew students from as far away as the Indian subcontinent until the Soviet takeover.

Arabic script was fairly widely used, specifically in the Persian alphabet adapted from Arabic, which was used to write Persian (aka Tajik) as well as local languages. These scripts were used until being replaced by Latin alphabets by the Soviets in the 1920s (these alphabets were then replaced with Cyrillic ones around 1940). Of course, in the mid-19th century, Central Asia was mostly agricultural, relatively poor, and had very low levels of literacy, so reading and writing in this alphabet (let alone another language) would be very limited. I don't have handy figures for the pre-colonial period, but just to give a sense of what we're talking about: by the end of the 1920s, adult male literacy in Uzbekistan was something like 27%, and that of women was 12.5%. Among Kazakhs in Kazakhstan it was less than 10%. And these figures were after the launch of Soviet "Likbez" campaigns to reduce literacy that decade, which indicates that prior to the Soviet period literacy rates would have been very low indeed.

But it's perhaps not fair to assume that low literacy in that period would equal low education. Much of the education was oral, based on face-to-face lessons between masters and pupils in religious madrasas (these were mostly male pupils and teachers, but it should be noted that Muslim religious education in Central Asia also held a place for female teachers known in Uzbek as otins. This type of reading and learning is, to steal an idea from French historian Roger Chartier, "intensive" rather than "extensive", the latter being a type of learning (often found in societies not using printed resources) where a society reads texts aloud, relies on memorization, and reads a few texts over and over, where the latter more commonly involves silent personal reading of a wide variety of texts.

In any case, the results of this religious education in Central Asia produced a paradox. A student could expect to learn the "Arabic sciences": grammar, prosody, and history, as well as the "rational" sciences of logic, in addition to learning shariat law. However, this learning was very much focused on learning and memorizing Muslim legal commentaries, and involved very little (if any) reading or learning of the Qur'an or Hadiths (one would learn to recite the Qur'an of course, but that didn't mean one actually learned what one was saying, let alone understanding the text or context). Thus while there was a system of education that included Arabic as one of its core subjects, it was a relatively elite education for the ulama, and extremely conservative, involving memorization and ironically little reference to the foundational Muslim texts written in Classical Arabic. Much of the rote memorization even involved learning mispronunciations of Arabic!

I should note that the above applies to Central Asian sedentary societies, not the nomadic ones surrounding them, where madrasas were nonexistent, and what Muslim practices and customs took place were largely through custom rather than through strong religious institutions (these areas were under tribal law - adat, not sharia law).

Some attempts were made to address the issue of low education and low literacy previous to the Soviet period, and this was mostly through the efforts of the Jadids, who in fact took their name from a new method devised of teaching the Arabic alphabet. The group was more or less founded by the Tatar Ismail Gasprinsky, but spread to Central Asia through the efforts of a number of other scholars, such as Mahmud Khodja Behbudiy. The group was reformist, although that term perhaps means something very different to 21st century Westerners than it would to the Jadids themselves - it was both modernist in that it wanted students to be taught subjects like modern science, geometry, history and arithmetic, and fundamentalist in that it wanted students to also understand Arabic well enough to read and understand the Qur'an and Hadiths without the centuries of added commentaries (in this, they were influenced directly by Salafists). To make a long story short, this new method made enemies, both among Russian colonial authorities and among the traditional ulama who saw their prestige challenged, and made the Jadids ironic allies of the Bolsheviks, before being ousted from power by the latter in the 1920s. Previous to 1914, one could find a range of books in such places as Samarkand that were printed in Tatar and Ottoman Turkish, but primarily in Persian (often these were printed and shipped from the Indian subcontinent) and Arabic, and on a wide variety of subjects: history, geography, sciences, medicine, religion, dictionaries, atlases, etc. However, the Arabic literacy needed for these works primarily came from the Jadid madrasas.

In short - among sedentary Central Asians, some knowledge of Arabic would be acquired by those receiving a rudimentary religious education, but this was mostly based on memorization, and thus overall knowledge of Arabic, especially previous to the Russian conquest, would have been both elite, and largely superficial.

Sources

Adeeb Khalid. The Politics of Muslim Cultural Reform: Jadidism in Central Asia and Islam After Communism.