Russian definitely changed quite a lot over the years. If we are using the range of 500-1500 CE as the definition of medieval then we can even say it only came into existence during that time, out of the Proto-Russian spoken in the eastern parts of Slavic society. If 16th century is the only concern, it is likely a conversation can be held between one of them and a modern time traveler.

First I want to give some background on the changes that were made through the life of the Russian language.

In 500 CE  the term “Slavic” was a broad term for languages spoken in the area now occupied by ex-Soviet bloc countries. By the end of the medieval era, 1500 CE-ish, we have something that is true Russian, distinctive from what would later become (or were already) languages such as Belarusian or Ukrainian.

The first significant changes between now and the time we are concerned with came under the tzars (1700s, specially Peter the Great). French became extremely popular amongst the wealthy parts of society, and many words used now are sourced from that language (этаж, meaning floor for starters, I’m sure I’ll think of more). This is very visible in much of the literature from the time, especially Pushkin who used untranslated French freqently in his writing. Also at this time standardization of the language became extensive as grammatical texts and dictionaries were published for the more wealthy populace. 

The Soviets simplified and standardized the language even more, removing the ѣ (yat’) character from the language entirely, though granted it had fallen out of common use earlier, as it became effectively identical to the е character.* It is important to note that many words simply substituted the two letters and were not changed in any other way, making it more likely the time traveler in question would be able to understand words affected by the change.

Post Soviet Union, a lot of English slang words have become commonplace (бакс, meaning buck as in currency, бро, literally just bro) and more technical jargon from asny period obviously wouldn’t be in your medieval persons vocabulary.

After that short history lesson, we need to divide the language into three parts: grammar, pronunciation and vocabulary.

Vocabulary is the actual words, such as “farm” or “store.” I don’t have any specific statistics on how many words are the same, because they most likely do not exist, but my personal experiences reading old documents as well as literature I would say that there are surprisingly many similarities between 16th century Russian. When reading old Church birth records or ship manifests, most everything is recognizable to me, a nonnative speaker. Granted, I do have a working knowledge of Ukrainian which helps in some aspects, but natives are still going to outclass me by miles, especially educated ones. It helps that Russian has a lot of reconizable roots and prefixes/suffixes, so words can be easier to deduce thre meaning of due to the recongizable parts together with context.

If the words are the same, then you have pronunciation (I’m including minor changes in spelling) would be another hurdle. Vowels have undergone the largest changes, with consonants remaining almost exactly the same. Spelling, especially use of the hard sign ъ has changed quite a bit, but is similar enough that with some effort that the spoken word would still be generally understandable. The same thing could be said of nouns in nominative case changing from a ье ending to a ие ending. 

Grammatically, it is remarkable how little has changed compared to English. Use of the vocative case was reduced to only some niches, but for the most part declinations are the same and sentence structure is irrelevant if the question is only basic understanding, as it is generally used to emphasize or deemphasize words.

Before I give the final verdict, I want to mention that it matters very much who the time traveler is, and who the poor person that he/she/they would be appearing to. A Russian serf from Ekaterinburg would be much harder for a modern person to understand than a muscovite politician, as the more influence a class has on literature, the more their own dialect will propagate into the future. Likewise, a modern native speaker from Ukraine would have their own quirks that might be confusing to someone from the past.

What all this boils down to is, yes, if a person from now and the sixteenth century had a basic conversation they would understand each other with some added time and effort. Words can always explained, grammar can be ignored or interpreted, and pronunciation can be slowed down and repeated. It would be very obvious they are not speaking the exact same language, but doable.

Please feel free to ask clarifying questions or for more detailed sources, I’ll do my best to find English ones, but I can promise little.

*Technically this was done under the provisional government of 1917, before the USSR was officially organized

Sources: Russian Studies Minor

https://nashasreda.ru/kratkaya-istoriya-russkogo-yazyka/

https://www.persee.fr/doc/slave_0080-2557_1982_num_54_3_5240 (no idea what the french is in there, it seems irrelevant)

I’m on mobile, so sorry for the inevitable issues with formatting.