How did the Catholic Church function logistically/liturgically prior to the invention of the printing press?

Ut In Omnibus Glorificetur Deus

I wish to take to somewhat different approach to this question than did Geremia and his excellent answer.

First of all, as the comment from curiousdannii makes allusion to the fact that in former days (Middle Ages and earlier) monks and nuns had great memories and that there was less active participation from the laity. Much of the laity could not read Latin and the priest and ministers generally read the various parts of the Mass, including the readings.

This was also a time to memorize, as there were not many printed texts of Scripture available. Monks and nuns did not have personal Bibles, but they did have the special memorization techniques and prodigious memories of medieval readers. It was also done in the group setting, “so that seeing one another they could encourage one another.” The aim was always to increase each brother’s awareness of God’s presence. Says Stewart, “a 14th-century Benedictine urged his readers to remember each evening’s common reading so that throughout the night, whether desiring sleep or prayer, they would have something to ruminate lest the devil find them at loose ends.” - But what did monks do all day? The holy routines of medieval monasticism

This is only part of the equation. Another point of interest comes from the Rule of St Benedict itself. St Benedict is known as the Father of Western Monasticism.

Before the thirteenth century, medieval manuscripts were initially produced in monasteries by monks working in the scriptorium, or writing room where books were made. More than five hundred monasteries existed in England alone by the twelfth century, and a typical monastic library might possess over three hundred books in its library. By the beginning of the thirteenth century, the growth of towns and the establishment of universities in Paris, Oxford, and Bologna led to the rise of secular scribes and artists who served students and professors as well as the nobility.

Later in the fourteenth century a rise in literacy and the development of an upper middle class created a large demand for illuminated manuscripts. The production of illuminated manuscripts of prayer books for personal devotions, called Books of Hours, increased dramatically. - About Medieval Manuscripts

Although St. Benedict's Rule obliged the monks to work for several hours a day. the whole Benedictine lifestyle was centered on the faith. Manuscripts were indeed expensive. but St Benedict was also a great promoter of the Catholic faith. We can see a glimpse of this in relation to this in Chapter 57 of the Rule. In the Life of St Benedict by Pope St Gregory the Great, we see can see this man of god freely giving things away, especially to the poor (Of the Life and Miracles of St. Benedict). As one could imagine, monks could have gifted poor churches the books they needed at a very reduced price or even completely free, that in all things God may be glorified (Ut In Omnibus Glorificetur Deus).

Of the craftsmen of the monastery

Should there be craftsmen in the monastery, let them exercise their crafts with all humility and reverence, if the Abbot so commands. But if one of them grows proud because of the knowledge of his craft, in that he seem to confer some benefit on the monastery, let such a one be taken away from this craft and not practice it again, unless perchance, after he has humbled himself, the Abbot may bid him resume it.

If any of the work of those craftsmen is to be sold, let those through whose hands the business is to be transacted see to it that they presume not to mingle into it any dishonesty. Let them be mindful of Ananias and Saphira, lest perchance they, and all who deal dishonestly with the goods of the monastery, should suffer in their souls the death which these incurred in the body. In setting the price of these things, let not the sin of avarice enter in; but let the goods always be sold somewhat cheaper than is done by men of the world, that in all things God may be glorified.

It could be noted that there are still monasteries that use the Extraordinary Form (Tridentine Rite) and one could still be amazed as to the ease in which these monks are able to recite those ancient texts of tradition by heart in Latin.