What is the name of the thing that the tongue does on the trī part in the word patrī? The word is at the 5:06 mark of this video! https://www.youtube.com/watch?v=OdQawsU2RFg&t=308s
As MPW said in their comment, this is a trill, a common feature of many languages. I'm no linguist, but I would classify the one in your recording as an alveolar trill, similar to that in modern Italian. The entry for Italian in the table of the Wikipedia article links to a recording of an example.
If I understand correctly what you are asking for (the "rolling" of the r on the front of the tongue in patri, librum, etc.), as mentioned previously in comments and answers, modern linguists would describe this as a voiced alveolar trill, which remains common in many modern Romance languages, e.g. strongly in Spanish, in Italian, in some dialects and registers of Portuguese, etc.
This is usually thought to be how Latin R was pronounced in most or all positions, which is commonly stated in many standard Latin textbooks, e.g. Wheelock's, etc. (although for a skeptical review of the available evidence, see Lucie Pultrova, "On the phonetic nature of the Latin R"). Provided that this is the correct pronunciation, ancient authors sometimes referred to the sound as "littera canīna" (the dog letter; e.g. in Persius, Satire 1.109-110) or compared it directly to a dog's snarl (Lucilius, qtd. in Aelius Donatus, Ter. Ad. 282; "Lucilius de littera R: irritata canis quam homo quam planius dictat").
There are many types of "r" sounds that phoneticists can distinguish. The one you have in the link is a "trill," a "tap," or a "flap." It is not clear to me what exactly distinguishes these three terms, and my brief survey of Wikipedia articles suggests that linguists don't completely agree either.
To me, each of these words requires at least one momentary contact between the tongue and some other part of the mouth that is briefer than what happens with a normal stop consonant. The word "trill" implies multiple vibrations, but I have seen descriptions of languages that talk about even a single vibration as a "trill." I think the words "tap" and "flap" imply a single contact.
Wikipedia describes "trills" as resulting from airflow, while "taps" and "flaps" result from muscular contractions.
This article on Wikipedia says:
Spanish features a good illustration of an alveolar tap, contrasting it with a trill: pero /ˈpeɾo/ "but" vs. perro /ˈpero/ "dog".
I think all Spanish speakers make this distinction when the sound occurs between two vowels, but not in other environments. In a word like Patricia, I think most Spanish speakers would use a single "flap" or "tap," rather than a trill, but the distinction would not be phonemic and might vary for different reasons. If someone used two vibrations, I don't think you would even notice. If they used three, you would just think they were trying to enunciate the word for some special reason. On the other hand, in the word, pero, described above, you generally have to keep it to one "tap," "flap," or "vibration" even if enunciating carefully.
Actually, I know of no language that has a phonemic difference between two types of R-sounds directly after a consonant. For example, French is often described as having an "uvular R," but in a word like "Patrice," I don't think anyone would actually use that pronunciation. It's hard to do a voiced trill right after an unvoiced stop consonant.
I hear the recording more as a "flap" or "tap" than a "trill," but that might only be a bias I am carrying over from the Spanish I speak.