All language changes so of course given enough temporal displacement there would be a lack of mutually intelligibility with any language and its precursor form. In the case of japonic, what might be comparable to for example modern Japanese is Late Old Japanese/Early Middle Japanese, which was spoken in about the same timeframe as what we define as Old English was spoken in. This is not to say that language changes at the same rate, mind you, but that this thought experiment needs somewhere to start. Some big differences between the two forms would be differences in the lexicon, morphology, phonology, and orthography. Some lexical changes would be that LOJ would have less sinitic vocabulary than MJ. The number of loans only increased as time went by in waves that are classified more or less by the dynasty reigning in China at the time: 呉音 "go on" (the Northern and Southern dynasties), 漢音 "kan on" (Tang dynasty), and 唐音 "tou on" (Song dynasty). LOJ depending on when exactly we're talking about would have the 呉音 and at least most of the 漢音 readings and loans. (It should be mentioned here that in some cases older readings of characters would get replaced by the newer readings from the new source. Being overshadowed instead of completely replaced was more common though). In addition, LOJ wouldn't have any of the 唐音 loans, which would include a lot of Buddhist terminology like 行脚 (angya "pilgrimage") and 和尚 (oshou "high priest"). It would also include a lot of common objects like 椅子 (isu "chair") and 行灯 (andon "paper lantern"). And finally there would be none of the western loans that came into Japanese from the Dutch and Portuguese, and then later from French, English, German, etc. That would also include common things that didn't exist in Japan at the time LOJ was spoken such as 天ぷら (tenpura "tenpura") which came about only because of Portuguese introduced cooking. You get the picture. This is only a cursory glance at lexical differences, but I hope it underscores that there would be issues in intelligibility.

What we now know as modern Japanese is the child of a dialect spoken in Edo (Modern Day Tokyo) in the Tokugawa Period. This dialect became common in Japan because of the Sankin Kotai system and its dissemination allowed it to achieve wide spread use by the end of the Tokugawa Period and the level of common language during the Meiji Period. This formed the nucleus of modern day Japanese. However, in addition to the gradual change of a language that adlerchen mentions the man from your television show would face a more difficult problem. Dialects in the Tokugawa Period could vary wildly. We know from written sources that even contemporary Japanese had trouble conversing if they went to a domain that had a dialect that was much different from their own. One samurai from Tosa remarked that when he first came to Edo he was so confounded by the speech of the people in the capital that he could understand less than half of what was being spoken. Although he notes he was able to figure it out quickly, many samurai had to take formal classes with teachers in order to understand the dialect spoken in the capital. This problem would be compounded the further you went back in time since prior to the Tokugawa Period there isn't enough significant urbanization to standardize language like what happened by the Meiji Period so dialectical differences would be even more pronounced. Even contemporary Japanese had language barriers so to expect a modern Japanese to be able to go back in time and fluently converse without any kind of issue is unrealistic.