I have an archaeologist who specializes in ancient ancient Northern European civilizations. Due to some handwavy mumbo jumbo, he is brought face to face with a supernatural being which still speaks fluent Proto-Celtic and/or Proto-Indo-European exactly as they were spoken thousands of years ago. Would the archaeologist be able to communicate verbally with this being or would the drift in accents over such a time frame be so severe that he could only communicate through writing?
First the bad news
Archaeologists are not historical linguists, and historical linguists are not archaeologists. It is not at all likely for an archaeologist to have more than an amateur level knowledge of Proto-Indo-European. Please note that Indo-Europenists are much very much rarer than archaeologists.
Neither Proto-Indo-European nor Proto-Celtic were written. (This is why archaeologists don't bother to learn Proto-Indo-European historical linguistics; it's not as if there is any hope of ever finding an inscription in Proto-Celtic or Proto-Indo-European.)
Nobody knows Proto-Celtic to any degree of accuracy.
We have a good idea of its phonemic inventory and of the sound changes which separate it from Proto-Indo-European.
We know some parts of its morphology.
We know some of its vocabulary.
We only know some parts of its morphology, and what we know is not 100% guaranteed to be correct.
We only know some of its vocabulary, and for those words which we know we are far from certain that we know exactly what they mean.
We have only a vague idea of its syntax.
Our knowledge of Proto-Indo-European is both a lot better and a bit worse than our knowledge of Proto-Celtic.
We have a good idea of its phonemic inventory, but we are still struggling to formulate a complete theory of how those phonemes were actually pronounced.
PIE had three series of dorsal stops, conventionally called palatovelars (written *ḱ, *ǵ, *ǵʰ in our reconstructions), labiovelars (written *kʷ, *gʷ, *gʷʰ in our reconstructions) and plain velars (written *k, *g, *gʰ in our reconstructions). It could be the case that they were pronounced as their conventional names and notation would suggest; it could be the case that the proponents of the glottalic theory are right and the actual pronunciation was quite different (involving ejective stops); or it could be the case that what we call palatovelars were pronounced as plain velars, and what we call plain velars were pronounced as uvular stops. We just don't know, and we don't really care all that much, because our reconstruction of PIE is intended more as a shorthand notation for the sound correspondence rules between the daughter languages.
PIE had three phonemes, conventionally called laryngeals and written *h₁, *h₂, *h₃ in our reconstructions. We know that they could function both as consonants and as syllabic nuclei, and we know the effect they had on nearby sounds. But we have a very vague idea of how they were pronounced; the best guesses are [h], [χ] and [ɣ] or [ɣʷ] on one hand, and [ʔ], [ʕ] and [ʕʷ] on the other. We just don't know, and we don't really care all that much, because our reconstruction of PIE is intended more as a shorthand notation for the sound correspondence rules between the daughter languages.
Our knowledge of PIE morphology may be significantly off. The basic problem is that it is quite likely the case that what we reconstruct for PIE morphology did never exist as a synchronic system, because our reconstruction may flatten long-term historical developments. (We reconstruct PIE by applying the comparative method to its daughter languages; those daughter languages separated from PIE over a timespan measured in millennia, and it's likely that the comparative method gives us a "compressed" view of PIE morphology.)
(To give a practical example, consider the English second person pronouns; looking at Modern English as a whole, we could say that English has three 2nd person pronouns: thou (singular), ye (plural) and you (singular and plural). But this synoptic view compresses centuries of evolution; such a system never existed as a common way of speaking.)
Some parts of PIE morphology, for example pronouns and demonstratives, are certainly poorly known, because the daughter languages are too divergent to allow us to make a solid reconstruction.
We know a lot of PIE roots, and we know how to make PIE words out of them. The problem is that in many (maybe even in most) cases our knowledge of the meaning of those roots is foggy at best. For example, we know that there was a root *ǵʰelh₃- in PIE; it was certainly a common word, because it has reflexes in all the Indo-European subfamilies; but we can only reconstruct its meaning as "yellow", or "green", or "to shine", or "to flourish", because its descendants include such disparate words as Latin holus (green vegetables) and helvus ("of the colour of honey"), Greek khlōros ("bright green") and kholē ("gall bladder"), English yellow and gold, and Russian zhyoltyy ("yellow") and zoloto ("gold").
(The good news is that for quite a few words we have rock-solid meanings; *ph₂tḗr meant "father", *méh₂tēr meant "mother", *bʰréh₂tēr meant "brother", *swésōr meant "sister" etc.)
We have very little good understanding of PIE syntax. The simple truth is that the descendant languages are all over the place, and just about the only thing we can say is that PIE has almost free word order, like Greek or Latin. Other than that...
And the good news
A historical linguist specializing in Proto-Indo-European (and there are not all that many of them, probably less than a hundred in all of Europe) would probably recognize the speech of the ancient being, and they would most likely begin to make sense of the being's utterances in a relatively short time, especially if the being is cooperative and engages in a bit of elicitation. (Show a rock, hopefully say pérwr̥; show his hand, hopefully say meh₂nos and so on.) The point is that we know the underlying system of PIE, and could probably fit a practical realization on our knowledge.
You have a more immediate question: why would you expect your archaeologist to even know Proto-Celtic or Proto-Indo-European? An archaeologist working with the Egyptians or the Sumerians or Old Norse or Ancient Greek or some other civilizations could be reasonably expected to know something of the language of the people they study because said people left written records as part of the archaeological record, so knowing the language is an added tool in their research.
That isn't the case the reconstructed languages; indeed, one of the reasons they're reconstructed in the first place is that there are no written records to use to figure them out. There's no benefit to a field researcher knowing that mel-nés (or something that sounds sorta like it) probably means "honey" in PIE since they'd never come across anything where knowing that was remotely important to their research, even if they uncovered an entire beekeeping industry from the time.
So unless the archaeologist also had a completely separate interest in linguistics, you need to explain that, even before you get to the problem of not being sure how it was really pronounced.
I agree with Alexander. There is a HUGE difference between understanding a written language and understanding the spoken version of that language.
Take Egyptian hieroglyphics. They are more than 3,000 years old. There are many people who can read them and have a general understanding of what is being described. However, there is no one on this planet that can accurately pronounce those words exactly as they were pronounced 3,000 years ago.
Because that language was not passed down exactly intact to anyone in modern times. And if someone claimed to know the ancient tongue, how could anyone possibly verify that?
To take a more modern example. The English language has evolved a lot over the last 500 years. It is unlikely that some one speaking only Old English would be able to communicate effectively with someone speaking Modern English.
And even in modern times. There are regions of the US where English speakers are partially or completely unintelligible to English speakers from other parts of the country. People from the South or the North East particularly have this issue.
When you add in regional slang and pronunciations, it is very unlikely that your hero would be able to speak to a monster that only speaks a 3,000 year old language.
For this to work, your archaeologist will have to be handwaved along a good friend who is a philologist, and in specific, an Indo-Europeanist who subspecialised in western languages (Italo-Celtic, Helladic, maybe Germanic).
PIE is thought to have been spoken up until about 2500 B.C., which is 4500 years ago. You're only looking for a daughter language. It's a good guess where any specific group was at any specific time, so your archaeologist might end up in the middle of nowhere.
Further considerations: Of course, PIE and PrCelt were not written down, though your philologist, should you choose to bring one along, can. Also, and most importantly, both of these languages are reconstructions. There are absolutely no written records of PIE, and Proto-Celtic is just tantalisingly out of reach of the earliest Celtic records. We do have some records of Gaulish and scraps of very early Irish.
By "reconstruction" we mean that modern, middle, and old languages are compared with one another --- their sounds, their words, their grammars, their syntax, and even some idioms --- and through this comparative method, ever older layers of language can be reconstructed. For example, we can see some kind of kinship between the words werewolf (Engl), vir (Latin), virá (Skt), fear (Irish), vyras (Lith.)
By looking at a whole mess of Germanic languages, we can deduce that the Primitive Germanic word was probably "weraz". We can look at all the other languages' data, and compare them with other similar words, and deduce that the Indo-European word was probably "wihrós".
It's almost a foregone conclusion that even the most proficient indoeuropeanist will not be able speak fluently with a native speaker of an ancient language. Unless that ancient language is Greek or Latin or Sanskrit, which we know well and what they sounded like. Anything older is educated guesswork.
Absolutely no. Not without extensive training.
Speech did not just happen. Humans did not just make up sounds, and form them into words. They used sounds they could already pronounce.
Babies babble phonemes of every known language, at birth. Over time, as they listen to their parents, they learn the important phonemes of the language of the parents, and stop babbling the phonemes that are not used in the language. Over time, they lose the ability to physically make the phonemes of languages they did not practice.
A case in point is the Chinese language. It uses almost no phonemes in the same way as  spoken by any Western language. Thus, not only is it very difficult for Western speakers to learn how to speak it, it is very difficult for Westerners to even understand the subtleties of the phonemes from the language. It takes a great deal of training our auditory system to pick up the nuances in the language. There are many, many words in Chinese that sound exactly the same to us, but there are subtle (to us) but pronounced (to them) differences in inflection that make it a totally different sounding word to them. They would proclaim that the two words don't even sound close to the same. Likewise, when a native Chinese speaker learns English, there are some English phonemes that they find exceedingly difficult to pronounce properly. There is absolutely nothing like them in their language, and they have therefore never developed the ability to form them.
Any language that was spoken thousands of years ago, would have been made from a unique subset of the phonemes that babies would have babbled at the time. We have absolutely no idea what those phonemes were, or even what they might be. There are no guarantees our vocal system would even be able to make them.
There is also no guarantee our auditory system would even be able to decode them.
So this archaeologist is in for an awful lot of speech training and speech therapy in a second language from this alien linguist.
See for instance https://files.eric.ed.gov/fulltext/EJ1152473.pdf
As a mono-syllabic language, each character has only one syllable and cannot be analyzed at the phoneme level as English is. (hong xu, 2014)
I haven't read all the answers in detail, but I think one aspect that isn't mentioned, is that proto-celtic or any other proto-language is an artifact that tries to sum up the traits that can plausibly explain the features as well as the differences we find in modern languages of a language-family. The actual, spoken languages at the time will almost certainly have been at least as diverse as what we observe today, at least if the speakers were spread out over any distances great enough to prevent frequent contact; just think about how dialects exist (and still evolve) in every moderne country, even with good communication media. The proto-languages, in my opinion, are in many ways what you would get if you tried to unify all modern, Indo-European languages into one, covering all variations in phonemes and grammar.
So, to answer you question: I don't think knowledge of a proto-language would be much actual use in practice.
It would potentially be very difficult. The written words of an extinct language would be preserved but how they are supposed to be used and pronounced in spoken language is mostly based on unspoken social contexts that a writer would assume any contemporary reader would know.
Take Latin for example. Despite being a nominally "dead" language for hundreds of years Latin was still used heavily in religious organizations and in the sciences. As a result, people still had a good handle on Latin rules of grammar and vocabulary. However, despite this the way they pronounced Latin looked nothing like how a Roman speaker would have pronounced it. A Latin speaker from 0 C.E. and 1900 C.E. would each sound like they were speaking complete gibberish to each other, despite speaking the same language. This is despite the fact that they could both potentially read a document written in Latin.
We only know this because of later efforts to reconstruct the pronunciation of Latin as to how it would be "originally" spoken (noting that Latin pronunciation probably evolved through the ages, as with any language), and there was a huge effort towards moving towards the "original" pronunciation from the Neo-Latin one at the turn of the 20th century. This was the crux of the "SIS-er-o versus KEE-kir-o" issue in Goodbye, Mr. Chips. Changes to Latin pronunciation are still going on even today as people continue to reconstruct how the language was pronounced back in the "old days".
And this is with potential context as to how to pronounce or translate certain words. With a language for which any potential context for pronunciation is long gone, as with Proto-Celtic or Proto-Indo-European, it would be a lot harder.