First we must see what kind of sources would mention antiquity in medieval times.

The most important ones would be the gospels. Even if the historical truth behind the Census of Quirinus is debatable today, it wasn't the case back then. Even if those texts are full of miracles etc., they still mention the name of Tiberius, the roman governor Pilate etc. Those knowledge would have at least been heard regularly during the mass, in addition to figurative representations in Stone sculptures, paintings, etc.

A second source close to it would be the hagiographic texts, lifes of the saints (hagiographies), list of saints celebrated each day with a short notice on their death (martyrology)... If the use of hagiograhic texts was very important, alongside the cult of the saints I can't really say if that influenced the knowledge of antiquity outside of the association of the pagans with persecutions of Christianity.

Further in the religious area we have the patristic litterature, the texts written by the fathers of the Church. A lot of it was originally written in greek though, that's why Augustine had a much greater influence on catholic than orthodox theology. One good exemple of informative document would be the letters of St Ambrosius of Milan. He spoke in it on his relation with Theodosius while the emperor was in northern Italy, it can (and still) show insights on the structure of the late roman empire.

To move to another subject we have ancient (roman) historical texts. Roman is a bit more correct due to the linguistic divide between the greek and latin world. First you must be aware of the varying number of manuscripts for those texts. In the early 9th Century a small revival in ancient knowledge is occuring under Charlemagne. More texts are copied, using a new writing: Carolingian minuscule script. That's good but they seem to have copied not the best manuscripts (with less mistakes), but those with a better cover, which can seem a like a good idea in front of a two copies of a text but is in fact a pretty bad idea. So in the 9th century appears the archetypes: versions of ancient texts that are the basis for all of the later copies (until the advent of papyrology). In addition some texts were populars and copied a lot but others were almost unknown. For exemple with have a lot of copies of the gallic wars of Julius Caesar or the letters of Cicero, used as an exemple of good latin or Suetonius used to associate the descirption of Augusus with that Charlemagne. On the contrary Tacitus, a very important roman historian is known by only one manuscript that also contains the all the known books of Apuleius.

As looking for various informations in various authors can be a difficult thing if you don't have a huge library and/or if you don't want to read 4 texts to have all the available information, new texts based on the ancient sources were written. The most notable that I'm aware of (note: I am not a specialist on medieval litterature) are the "Faits des Romains" (acts of the romans). A biography of Caesar based on different ancient texts about him. It was very popular at the Court of Burgundy in the 15th century.

It's also remarquable that a great number of mythical history of the middle age want to connect the origin of important cities to Ancient glory. Rheims was supposedly founded by Remus, the heirs of Hector (like another Aeneus) founded Paris etc.

Then there are monuments and landmarks. Many theaters and amphitheaters of the roman world became inhabited, and honestly I don't know if the inhabitants knew where they really were living. Note that the abandonned parts of Rome changed radically. The forum became the Campe Vacino (cow field) and the Palatine the goat mount. The catacombs on the other hands had a strong connexion with the Church and the aforemention texts, the traditions around them (and the commerce of relics) was quite strong. I also remember a book on proto archaeology that mention a priest that is called because peasants founded a statue in a field. He had the knowledge to identify the statue as the pagan god Mars and to make a little theological comment on why he shouldn't be worshipped.

Rome also left traces on the legislative level, the commune of Rome still had a Senate, well aware of his origin even if their power were restricted over time. On a similar subject, administrative name in towns of Italy or southern France were for centuries still very close to their ancient counterparts, they were most of the time ruled by Consuls. Note that the name was not applied to a city governor in roman time. It was the lack of a higher power that opened that possibility.

In Germany, well in the "Holy Roman Empire" literrary tradition also had an influence on politics in order to create arguments in favor of the emperor against papal supremacy. That's in this context that the "donation of Constantine" of the western empire to the church should be placed. The emperor Barbarossa on the other hand used roman law to show his superior position to the Pope.

Small edit: be aware that there is a life of Alexander the Great in latin by Quintus Curtius.

So that's all for now. It doesn't cover all the subject, but that's better because I know I don't have the knowledge and the library to do it entirely. You can ask for more specific questions of course.