This issue, as usual for the field of historical linguistics, is a bit complex and actually contains of three parts that were addressed in the question above. First, did Polish language remained unchanged over the lat 780 years (I assume the 'Mongol assault' is the First Mongol Incursion that culminated in the battle of Legnica in 1241). Second, would the Polish language in 13th century be mutually intelligible with that of 20th century. And third, would an user of modern Polish be able to easily communicate with his ancestors back then. In short, the answers are 'no', 'largely yes' and 'it depends'.

The main difficulty to provide a quality analysis of the Old Polish language is the relative scarcity of the contemporary sources, that are far less numerous that that written in Old English or Old High German. There are basically no known texts written in Polish that would predate 13th century and there are only few from 13th century, all of them being ecclesiastical and liturgical texts. It is largely agreed that the oldest inscription in Old Polish is a short sentence in an otherwise Latin source Liber fundationis claustri sanctae Mariae Virginis in Heinrichow ('Book on foundation of the Holy Virgin Mary cloister in Henryków') that has been written in 1268 (and said phrase might not be representative of contemporary language as it was written using only Latin letters that lack many of the sounds specific to Slavic languages). So, as far as the history of Polish language is concerned, we do not exactly know what the language used in 1241 looked like and all conjectures are based on sources from the very late 13th and 14th century, such as 'Bald Mountain Sermons' (named after the Benedictine abbey on the Bald Mountain where the text has been found), Sankt Florian Psalter or mid-15th century Suled Codex containing legal acts from the times and various judicial statements and confessions that became to be widely recorded in Polish rather than Latin since late 14th century.

The first issue mentioned above is related to the evolution of the Polish language. Although Old Polish was not as widely different from modern Polish as, for example, Old English differs from Modern English heavily influenced by e.g. Middle French, there are still many differences that make it very different, if largely intelligible from modern language. These differences, however, are not alien for a modern speaker thanks to quasi-historical stylization popular in literature and movies that, although not really accurate from historical standpoint is close enough to provide rough approximation of the Old Polish. In other words, the language differs quite distinctly, but these differences are not severe enough to prevent understanding. The differences in declension and pronunciation are more similar to those of a different dialect rather than separate language and thus, although hard to mimic properly in speech without getting accustomed to them, would most likely not make understanding harder.

The second issue is heavily dependent on the speaker in question. I haven't read this particular novel and don't know who the protagonist is, but it s generally agreed that the differences between West Slavic languages in 13th century were smaller than nowadays (e.g. early 15th century Hussite song Ktoz jsu bozi boyownici can be largely understood by modern Poles) and thus a Polish native speaker who also knows modern Slovak or, to lesser extent, also Czech, would find Medieval Polish easier to comprehend, especially in southern part of the country.

The third issue, although similar to the second one, is also dependent on the speaker and type of communication. In general, passive fluency develops faster that the active one, i.e. a person learning foreign language can understand written or spoken text better than they can speak or write similarly complex expressions. This is especially visible when such person already knows another language from the same family and can deduce meaning of the words using their knowledge (of course, this is where 'false friends' some in question, but these are rare and can be avoided when the context is clear). This means that a modern Polish native speaker, preferably with some fluency in Slovak would find Old Polish rather intelligible but could find speaking in this language rather difficult before getting used to it. I don't say they would have been not understood, but the initial fluency would be far from perfect. On the other hand, this would not necessarily make communication difficult, as regional differences between Polish dialects in 14th and 15th century could have been rather extensive and thus a time traveler could have been easily taken for someone from other part of the country.

So, to sum it up, Old Polish used in mid-13th century was distinctly different from modern Polish, but the differences were not that great to hamper mutual understanding to a significant degree.