The short answer is yes (this is really a linguistic question rather than a historical question, but alongside Islamic History and Modern Middle Eastern History, I studied Arabic; I have a Master's degree in Arabic).

There are two main 'registers' of Arabic: a formal, written register known as 'fussha' (literally 'clear') and an informal colloquial register known as 'ammiyyah' ('illiterate') [EDIT: not 'ummiyyah', as I had it previously].

Ummiyyah Arabic varies from country to country, and although mostly mutually intelligible, there can be substantial differences in vocabulary and grammar between different varieties (e.g. Moroccan Arabic and Gulf Arabic). These are probably greater than the differences between American English and British English, but a little less than the differences between, say, Spanish and Italian.

Fussha Arabic is not divided regionally (except in some minor matters of pronunciation), but is divided into two classes: Modern Standard Arabic and Classical Arabic. Modern Standard Arabic (MSA) is the type of Arabic you'd hear on the news, or read in a respectable newspaper, in any Arabic-speaking country. Classical Arabic is the Arabic of the Quran and classical Arabic literature.

Now to your question: because of the cultural influence and religious value of the Qu'ran, which still serves as the basis for 'correct' Arabic use, Modern Standard Arabic has diverged from Classical Arabic much less than today's English has diverged even from the English of the King James Bible. Anyone who can read and speak Modern Standard Arabic can also read and speak Classical Arabic.

So any educated Arabic speaker today could read something from 1200AD more or less as if it were written yesterday. They would also be able to make themselves understood, and to understand. The only real difficulty would come in the use of occasional modern words and expressions, which an educated speaker could easily avoid in favour of more Qu'ranic expressions.

It would be much harder for a non-educated speaker. If someone only spoke ummiyyah Arabic, and had no foundation in Qu'ranic Arabic at all, they would probably struggle.

Interestingly, a very similar question came by a few days ago on Twitter.

The short answer is: it depends.

The longer answer is: it depends on the Arabic speakers' place of origin, their degree of education, and (obviously) the topic of the discussion. Let's go through these issues in reverse order.

Topic of discussion
Obviously, a speaker of Arabic (or any language, for that matter) living in the 13th century is probably not going to understand what you're talking about if you start mentioning things like computers, smartphones, memes, and what not.

But let's assume that you're sticking to things that they may be more familiar with. In this case perhaps a topic like pre-Islamic poetry, Arabic grammar, Islamic law or Qur'ānic exegesis are very likely topics that you might be able to discuss with some degree of success. As these are very classical topics by nature, the vocabulary used to talk about them hasn't really changed significantly between the 10th century and today. Congratulations! But there are a few more obstacles before you and your envisioned Usama b. Munqidh will be able to have a meaningful conversation.

Degree of education
Another important obstacle to whether you and your newfound 12th century Arab friend are able to hold a meaningful conversation depends on your respective degrees of education. As you may imagine, the Arabic-speaking world is an immensely vast and linguistically diverse area. It is estimated that are at least 300 million native speakers of Arabic and over 200 million speakers who learn Arabic at a later stage.

But there is a kind of linguistic situation, which although not really unique to the Arabic-speaking world, is very characteristic. This situation is known as diglossia, a term that Charles A. Ferguson popularized in his eponymous 1952 article. In the Arabic-speaking world, there are two main varieties of the language that exist side-by-side and are used in different quotidian settings: the L (low) variety is the language you speak at home with your parents, with your friends on the street and when going shopping. The H (high) variety is the language you may (used to) hear on TV and the radio, at university, and whenever the president or monarch gives a speech.

To a degree, this situation exists in virtually all languages: the way we speak at home is never going to be the exact same as the one we use in more formal settings. However, the morphological, syntactical, and lexical differences between the "high" variety of Arabic and the Arabic dialects are often so large that they are not mutually intelligible, or only to a very limited degree. Unless the speaker of Arabic has attended school and acquired Standard Arabic (which is not spoken natively by anyone) they are going to encounter serious difficulties in trying to communicate with Arabic speakers from other countries; the further away, the more difficulties they are likely to encounter. But let's assume that you, our magically transplanted 21st century Arabic speaker did very well at school and passed all your Standard Arabic exams with flying colors. When you find yourself in front of Ibn Khaldūn all of a sudden, after passing through the initial shock, the two of you might be able to just communicate with one another. But if you and Ibn Khaldūn decide to go out and sit by a tea- or coffee-house in Tunisia, Egypt or Syria, you might find it hard to make yourself understandable to the (likely) uneducated people roaming the streets of Tunis, Cairo, or Damascus. Which brings me to the next point:

Country of origin.

As I mentioned previously, the Arabic world is vast; without education, a speaker of one of Morocco's dialects may find it near impossible to make himself understandable to a speaker from Basra. And even if the latter got the point, the response may be (near-) unintelligible gibberish to to the former. So if you will want to be able to talk to the hoi polloi in the city or cities you suddenly ended up in, you have to account for the following as well.

All languages go through a process called linguistic drift: the more languages spread, the more they tend to become different. This is entirely natural: think of the English word church and compare it to German Kirche or Swedish kirken, which are all pronounced differently. Such sound changes are just one aspect of linguistic change. With regards to the spoken Arabic dialects, there are certain very visible sound changes (a very common example in North African Arabic variants is that short vowels are dropped), but there are also significant changes with things such as syntax (many Arabic dialects have a different way of indicating possession than Standard Arabic) or morphology (loss of gender distinction in some verbs, for example) and words.

What you should note, though, is that despite this huge linguistic variety, Arabic has an advantage similar to that of Hebrew. As you may know, Hebrew is the only real example of a "dead" language to have been successfully revitalized (it's actually a bit more complicated, because Hebrew was never really dead, but that's another topic), and that's mostly due to the fact many people knew Hebrew as a liturgical language. In the case of Arabic, many, many people consider it a holy language, which also means there's a certain constant pressure Standard Arabic exerts on the spoken language.

Conclusion

So if you account for all of the previous potential hoops and ladders it's certainly not impossible that you and a similarly highly-educated Arabic literary man or woman might be able to have a conversation. Just try not to confuse them too much by talking about your smartphone or the internet or satellites, try to dose the input.

Important postscript

This all being said, there are some pretty important caveats and additions I would like to mention.

You may have noticed that in my answer I mostly used the term Standard Arabic. This is the most neutral term I could consider, but this is already a measure of some controversy. In Arabic, there is only one term: al-fuṣḥā, practically meaning "clear" or "pure [Arabic]". Depending on the region, there are different terms used for vernacular Arabic: dāriǧa, ʿāmiyya, laḥǧa, all meaning something like "vernacular".

However, outside the Arabic tradition you will find words like Qur'ānic Arabic, Classical Arabic, and Modern Standard Arabic. There exists the idea that these are practically the same thing, but it's a bit more complicated than that. Firstly, there is some extremely interesting recent research on the linguistic nature of the Qur'ān which indicates there are some significant differences between the Arabic of the Qur'ān and the Classical Arabic of the medieval period. Examples include the likely existence of four vowels instead of three and the loss of the glottal stop. If you're interested in this, I suggest checking out the work of Marijn van Putten.

Secondly, Classical Arabic and Modern Standard Arabic are very similar, but not entirely the same. In the late 18th and 19th centuries the Arabic literary world went through something of a renaissance, called the Nahḍā, which means...well, "renaissance". Anyway, during this period the literary was revised, people got rid of some of the more arcane features of Classical Arabic (including a rarely-used energetic mood, some changes with regards to syntax). As you can imagine, these topics are pretty controversial and are always good for causing a lively discussion.

Sources

Ferguson, Charles A. 1959. “Diglossia.” WORD 15(2):325–40. doi: 10.1080/00437956.1959.11659702.

Van Putten, M. 2018. “Hamzah in the Quranic Consonantal Text.” ORIENTALIA 87:93–120.

Versteegh, K. 2014. The Arabic Language. Amsterdam: John Hopkins.