The trouble with this question at least from how I understand is that Japanese has historically had different styles in different contexts. As a basic example, "basic" Japanese uses the verbs "Iku" (to go) and Kuru (to come). However, "business" or "more formal" Japanese uses "irrasharu" for both, the verb itself being used in very polite contexts.

For that matter, the imperial court spoke Japanese very different from the common folk. For example, when the Emperor Showa (Hirohito) announced Japan's surrender in WWII, many people did not understand what he was saying as he was using an extremely archaic form of Japanese.

Even more confusing is the fact that people in different parts of Japan speak totally different dialects, the difference being more pronounced prior to the Meiji restoration. For example, in Kanto (Tokyo area), women may say "Watashi" to mean "I," while in Kansai (Osaka area), women may say "Uchi." For that matter, Kyushu-ben, or the Kyushu dialect, is extremely different from standard Japanese.

Having said that, one of the first Japanese novels, the Tale of Genji, came out at some time in the 1000s-1100s. In its original form, which is Heian era, imperial court Japanese, your typical Japanese speaker will be very confused very quickly. The grammar is different, the words are all written in hiragana (as opposed to kanji, or Chinese characters), and it is imperial court language on top of all that. Needless to say, it would be extremely unintelligible. For the most part, the "sounds" of the words would be similar. When I say sounds, I mean each "kana," or syllable, of words. For example, Tokyo has the kana "Tou" and "Kyou." Those particular kana would be pronounced the same in both ancient times as well as today. However, certain kana were used back then that are not used today (ex. Wi, Wu, We(h), Ye, and Yi). A Japanese person today would think you are either way out of your time period, or that you are speaking gibberish.

Sources:

de Bary, Sources of Japanese Tradition

Tale of the Genji (Genji Monogatari, Tyler's version)

Interestingly enough though, Japanese children have classes where they extensively study classical Japanese starting from middle school. This subject is called 古文 or kobun in Japanese.

In general, it is required for them to be able to read and understand the classics of Japan. They are also often required to at least try to write some poetry of their own to show their command and mastery of the prose they study and they also hear the kobun teacher read passages as they would have hundreds of years ago.

Even outside of school, there are often references to the classics, whether it's quoting on variety and talk shows, as questions on game shows, or used in comedy acts. Kobun is pretty ubiquitous in Japanese culture, whether in school or out.

It's mandatory to learn as a part of standardized curriculum across the nation and it's also often included on highschool and college entrance exams so most kids are forced to study and become at least acquainted with the prose that is used in classical literature dating as far back as the Heian Period, which was over 1000 years ago...or risk bombing an entire part of the exam.

This means that Japanese students are generally much more well acquainted with classical Japanese than the average student is acquainted with Middle English in, let's say, America. Er, assuming they study properly.

This does mean that in general, they have a better chance at communicating if suddenly forced to speak with the average Japanese person plucked from 1114. If someone was particularly good in their kobun class, they'd probably have excellent chances.

The real problem actually begins the farther away you get from 1114 and ironically, closer to modern day. You see, the classical Japanese that's used in literature that modern Japanese students study in kobun has been used as a form of prose much longer than it was used for speaking verbally.

For example, people were using the same forms and styles to write literature as late as the Meiji Period and World War II, while they definitely did not speak that way to people around them.

By the time it was the Sengoku Period (16th century), plenty of people would certainly have difficulty understanding the prose without an indepth education as modern day people receive in regular education.

Another problem is that uneducated people would also have problems understand certain forms, grammar and vocabulary. Illiterate people's general range of vocabulary would be smaller and more direct. If a modern day person spoke in perfect literary prose, it would sound very weird even when it was a contemporary thing.

Other problems, as ParkSungJun said, are the various dialects. If anything, dialects in 1114 Japan would have been even more disparate and dissimilar to one another than they are today.

Also, ParkSungJun raised a great point about the Imperial family and nobles. They would speak differently when court was in formal session and always when addressing the Emperor directly. This was part tradition, part religious and spiritual reverence, and part politics.

It is theorized that it was very politically advantageous that the Emperor could not directly communicate with people who were not already a part of the power circles, making it easier to sequester or do away with an Emperor who would not cooperate with the aristocracy or, later on, the Shogunate.

All in all, the average Japanese person probably stands a much better chance at communicating fully than an average English speaker does, if for no other reason than they had a class based around classical Japanese as well as it being a test subject to be able to get into competitive highschools and colleges (As far as I know the SATs do not require knowledge of Middle English). Outside of that, culturally, classical Japanese is much more common than Middle English is in modern media.

Yes. In addition to points already made, Japanese used to have a number of different dialects that weren't mutually unintelligible but were pretty close. These only really ceased in Generation X Japanese who more or less spontaneously all chose to start speaking the Tokyo dialect (the most notable holdouts are Yokohama and Okinawa, but even in those two places there aren't many young people speaking their parents's dialect).

Ancient Japanese is studied as a foreign language by modern Japanese. Most of the focus is on reading it. Pronunciation would have shifted massively in that time and a modern Japanese person would be totally unable to understand a speaker from 1100 apart from an occasional word here and there.