If a well-educated man or woman (master's degree level of education) with 20 years of experience in a field of your choosing was hurled back in time 2000 or so years, how could they most influence the progress of technology and industrialization with the knowledge they take with them? For the sake of argument assume the time traveler can speak the local language(s).
Write books. More precisely co-write books with a local experts to ensure they can be understood in local context.
The beauty of literacy is that it allows people to share the knowledge and experience in a way that is almost infinitely scalable. Even before printing press, which you could introduce, spread of information was limited by economics of who can get the books and spend the time to study.
Let's assume that all your readers are unrealistically dumb and can only use what you write one thousandth as well as you could. It would still only take a thousand people to outperform what you could do yourself. Something that in theory could happen within a decade of writing the book. And, in practice, locals would be better than you in adapting your knowledge to actual circumstances. So it probably would take less than a dozen people to outperform you working personally.
Another way to to get the same result is to note that since the time traveller has a unique quality, the possession of future science, (1) time spent on practical things locals can do is wasted and (2) your absolute priority must be duplication and preserving that unique resource, nothing else can be as valuable a contribution. (Unless you have another unique talent.)
While I already from the beginning noted that you can introduce the printing press, the fact that both comments specifically noted it despite me already having written it made me realize that it is kind of important point. If you rely on writing books and then letting natives distribute them then obviously anything you do to boost that distribution process amplifies your effect.
So I'll point that the printing press is not the only thing you can introduce and not necessarily even the best. There are some earlier inventions that had significant effect on the distribution of knowledge.
The alphabet made learning to read and write easier and as such was an enabling factor in making the printing press matter. This was already used in the west well before the time specified, but might be useful to introduce depending where the time traveller ends up. Many languages had long periods of using imported writing systems not very well suited to the language.
Paper is lot cheaper than the materials preceding it and if unknown would be a major contribution in making dissemination of information cheaper and hence faster and wider.
Codex was a major improvement in density and robustness of information over scrolls. It was invented by the Romans, so it might already be available when the time traveller arrives, but if not it should be introduced. In fact of all the technologies here it is the one that is most value for buck and there is no plausible reason not to make it available if it isn't.
I am not saying a doctor here, as doctors have a more mechanical job than nurses. It is nurses who physically deal with patients. A nurse with 20 years of experience would basically revolutionize how patients are taken care of, back 2000 years. Most important changes would be:
Hygienic conditions of the patient. Back uptil the medieval ages, there was practically zero awareness about hygiene and health. A nurse with modern knowledge about hygiene would completely turn the things around in this field and their heal-rate would be immensely higher than other medicine men.
Food of the patient and generally healthy diet. This one should be self explanatory.
Social and environmental hygiene. For example, no stagnant ponds = no mosquitoes = no malaria/yellow fever. Boil milk before consuming. Etc, etc.
Care for the patient. We now know far more detail about human anatomy and physiology than what was known 2000 years ago. This includes keeping a bleeding wound elevated, ways of decreasing pain for a bone breakage etc.
Bleeding stoppage and first aid management. Self explanatory.
While a pharmacist would not be able to manufacture the pure drugs he/she could in modern world, he would still be able to administer the source of naturally occurring drugs for ailments.
Using cinchona bark for treating malaria patients. This was unknown till medieval times. This is the only example that comes to my mind. A professional pharmacist would know far more examples.
They would also be able to tell (for a lot of diseases) which things not to administer. In times when disease was thought to be caused by ghosts and bad spirits, a lot of potentially lethal remedies were carried out for treating diseases.
They would also be able to prepare extracts from a lot of natural sources of medicines.
A Major Rank (Or Above) Military Officer
Now this doesn't involve masters level education but military training and education is definitely something not everybody knows. The revolutions in this sector would be:
Management and training of an army. Back 2000 years, there was very little military training. Most of the times, the kings kept a small formal army and when raiding or defending, they would enlist a huge number of irregulars as militia for the job. A seasoned military officer would put his troopers through regular exercising and training, massively improving their output in the battlefield.
Empty hand and close quarters combat. Self explanatory.
Guerrilla warfare principles and tactics. This would be an entire new form of warfare for those people and it would potentially be able to change history on a large scale.
Biochemical warfare. Yes, crude forms of biological warfare could be carried out even 2000 years ago. Refer to this book by Adrianne Mayor for reference.
Scorched Earth Policy. A very sinister form of inhumane warfare that is/was very effective too. A modern military tactician would know this field of warfare in far more detail than the ancient peoples.
Importance of army maneuverability tactics. Circling and cornering the enemy.
Production of primitive types of canons (provided that saltpeter was available in a mixture or pure form). These would have the potential to turn the outcome of battles easily.
Civilian training to counter an invading army. This one would have very important consequences and might be able to rewrite history.
A Civil Engineer
Although civil engineers work with modern technology, the very basics and principles of this field can be applied in any time and environment.
Longer lasting, better bridges.
Dam building and management.
City Planning And Management
This one would also have highly visible and practical impacts.
Deciding where cities should be built, considering the terrain, populace and sources of necessary items (water, firewood etc).
Planning the city accordingly. Distributing residential and commercial areas. Street planning. Defensive planning.
These are only some of the vast variety of experts which could/would revolutionize life in their respective fields back 2000 years. In fact every expert of his/her field (except the ones which are extremely technology dependent e.g. aeronautical engineers, space scientists, theoretical physicists etc) would bring major changes in the community where they end up at, 2000 years ago.
Given enough money/authority, one nurse from today's times would be able to cure more patients than 50 other medicine men. A 50 men rigorously trained army under a modern military strategist would be able to virtually obliterate or otherwise severely damage an army of 5000 militia trained in ancient ways. Practically such a strategist would be able to make his country invincible against a 5 times greater threat any time, provided he had the highest rank in military and was sufficiently provided for, for his needs, by the ruler/king. One civil engineer would be able to connect regions and places otherwise disconnected completely. He would also be able to build roads to increase trade caravans speeds etc. One city planner would build far more successful and healthy cities than an unlearned man with 5 times the money to spend.
Knowledge is power!
What can make a difference and is readily achievable (once the time traveller manages to speak with the right people) is to introduce positional numeral notation, along with multiplication tables and pencil-and-paper multiplication and division algorithms. Especially division of bigger numbers had been rather tedious before.
Another simple invention is optical lens - especially in telescopes. Does require some glassmaking skills and basic knowledge about optics, but not much more.
These are of course just easily reached starting points, whether they would leave to any progress depends on many external factors.
No degrees required in either case.
Honestly? not much. One or two of the right degrees might make a small difference in a small kingdom somewhere, maybe be lucky enough to bring a golden age. But most degrees are designed to give you knowledge for interacting with today's technology and societies. My Master's degree is in Information and Technology. It would be completely useless more than about 100 years ago.
History would and maybe psychology would be two of the more useful degrees, especially if you were an expert in what ever setting you get sent back too.
I would guess architects and engineers might be useful depending on when and where they get sent. Most engineers don't know how to smelt iron or make high quality steel.
My other hobbies and skills would be much applicable. I do a bit of wood working and black smithing which might hold me as a not entirely useless person. If people would believe me, I might be able to introduce scientific principles and theorems if I could convince anyone to listen. But really for most of us, even with a useful degree, we would be unlikely to have enough influence to change much, unless we became an adviser to an ambitious leader giving advice and using what knowledge we have to help change the course of history. Alexander the Great, Julius Ceasar, Ghengis Kahn etc.
I think it probally depends on what their area of expertise is.
Though I think I would start with something in the physics realms. I remember Richard Feynmanns answer to a similar question about what peice of knowledge was the most important.
He said simply "Everything is made of atoms."
A material scientist must be specalized in carbon nano-tubes, but he usually buys his tools from an engineering company that specializes in steel and manufactoring, ect. Your basic quality of life requires millions of people working in concert to produce anything. The term is often mis-used in political debates, but this is the actual 'invisible' hand that Adam Smith wrote about in his book "Wealth of Nations".
Ultimately the invisible hand is not operating with that level of complexity in a pre-industrial socieity, no one person can do much for the world even with advanced knowledge of the sciences. They would need to rediscover arts and crafts that don't even exist in todays day an age. You would need to spend your days mastering the intricate craft work of lens or metallury in order to do anything. No google either, so you have to learn the hard way, by tinkering. You would be a bit better off than most people trying to develop new technology, because you know what is possible, but your ignorance on the path to get there would be appauling.
I think the most impact could be made through some kind of engineering though. Like building a working printing press. You have metal, you have labor, and with time to learn the trades you need and the abstract knowledge of the rules of thermodynamics (which are yet undiscovered), it's a doable project. This could create wealth in the local currency, earn you laurels from well connected landed gentry, which you could then use for future investments into technological research and inspire a new generation of technologists to come up from behind you.
On your death bed, you leave a book with all you can remember about your western education. Your word becomes a trove for an entire generation of researchers to mine and then apply the available technology of the day, to develop the technology you describe, along principles you can elucidate. Assuming of course your a science major with a fairly broad expertise in a wide viarity of subjects. Specialists would potentially produce less value than someone who is able to communicate great ideas and understand abstract principles.
The name of the book in question? "Everything is made of atoms"
I think Burki had a very valid point. Namely, that the infrastructure is not in place to support almost anything. Nursing is a good, and creative, answer, but the only thing a nurse could practically do is convince people to wash more, resulting in possibly longer life-spans. But that doesn't kick-start any technological revolution.
To begin, there are a few serious issues to consider for such a traveler going back 2000 years and wishing to kick-start the industrial revolution.
First off, you are arriving shortly before the collapse of the west. The Roman and Egyptian empires have already begun their 300-400 year collapses and I would suggest that technological innovation would do little to slow the rot at the core of those empires (although I might be wrong. Perhaps it would allow expansion to the Americas much earlier which could have a knock-on affect for their societies... After all, the ancient Greeks, and by extension almost certainly the Romans, already knew the world was round).
Secondly, we are in an iron-age infrastructure (in Europe) and any advances will need to be made within that framework (check https://en.wikipedia.org/wiki/List_of_archaeological_periods for a comprehensive list of periods around the world).
Finally, it's helpful if the traveler has some decent knowledge of engineering/physics or such that can aid them in devising/crafting iron/steel contraptions, and an healthy interest in the history of science so they understand the scientific evolution of technology.
So, it will be important where the traveler is and where they can get to in order to be most effective. They will want to be where there is iron forging (and preferably good steel production although they could possibly offer improvements in techniques here) in order to craft items that can be used for steam-powered devices. With fairly rudimentary knowledge and some skilled smiting, smelting, and forging, they could possibly build many fundamental devices from the early industrial age... If they could get the quality and quantity of resources needed. This would undoubtedly mean gaining a favourable ear with someone in power, and would almost certainly mean they would have to start in Europe (or possibly China), with the knowledge that they would have to develop both the technology and find a way to distribute both the tech and knowledge so that it remained after the social collapses (and in the face of the coming wave of uber religion - the rise of the Catholic Church).
This topic has been discussed a lot around here, in many interesting facettes. The general consensus seems to be that no matter how good you are at what you do, jump back two millenia and you have practically no ground to stand on.
Every little bit of technology you typically use to do your job has not been invented yet.
To make that really clear: You cannot even get a pencil anywhere!
So, I doubt that being an expert in any field will help you very much. One of the main problems of the time is that people were short-lived and very superstitious. The one thing, if there is any one individual thing, that started the industrial revolution, was enlightenment, or, more precisely, a society that had (more or less) stopped burning sceptical thinkers at the stake.
Now, if you fond a profession that could jumpstart this kind of mindset, i am sure that would be the biggest lever you could find.
In the mean time you might want to try to teach them a few things about hygiene (without being called a witch or sorcerer) and make a large number of very small changes wherever convenient.
Oh, and don't forget to find a way to ensure that the people from the next village aren't going to accuse the whole of the village of witchcraft or such, and be it only out of jealousy.
I think the time traveler's profession is much less important than the ideas he can bring (though if I had to pick a profession, I'd say mechanical engineer with a familiarity in early industrial tech would be best, or a chemist):
Idea #1: Intellectual Property
The idea of patents was crucial to the industrial revolution. To wit: that someone who comes up with a new idea, shares it, has a monopoly on making money off of it for 20 years. If anyone wants to use it, they have to license it from the inventor. This is a major drive for the ramp up of technology we've seen from the beginning of the industrial revolution til now. Note, that Heron of Alexandria made basic steam engines and wind powered machinery over 2000 years ago. Did anything come of it? No. Why? Because no one was operating in the environment where it you invent something useful and bring it to market, you can potentially become fabulously wealthy. The only other major driver of technology is war.
Idea #2: Titles and Property Rights
As in property titles, not aristocratic titles. Having an authoritative titling system, and a government that fairly enforces property rights, is key for the kind of capital buildup and capital utilization you need in an industrial revolution.
Idea #3: Checks
As in bank checks, and the system of their settlement between the banks themselves. Being able to securely and easily transfer large sums of money this way really unleashes commerce and capital development. Of course you need the concept of a bank then too.
Idea #4: Germ Theory
Understanding that many diseases are spread by germs will save many lives and help avoid some of the nastier pandemics. So, your human resources are available longer and more stably.
So, in a nutshell, the thing to do is not necessarily to provide the tech of the industrial revolution yourself, per se, but to use the lessons of history we've learned to create the environment in which the industrial revolution can happen.
That being true, perhaps the best profession to send back is an experienced politician, since you'd need to convince the local government to implement most of this.
Disclaimer: I am a CS major; non of my education would be directly applicable. Nevertheless, there are few things I could do in those times:
is a necessary precursor to industrial revolution. Introduce crop rotation and Dutch plough. Invent sugar extraction from beets.
Invent oil paint and primed canvasses. Invent paper. Invent etching and printing.
Distillation and aging of spirits
Hot air balloon; boomerang and frisbee; mechanical clock.
L. Sprague DeCamp's "Lest Darkness Fall" - can't believe it hasn't been mentioned, since it's almost exactly the same idea. His solution was introducing moveable-type printing (and yes, developing the necessary precursors was an issue). First, he made a name for himself with positional notation and double-entry bookkeeping (Impressing mathematicians doesn't get you much. Helping rich people get richer, on the other hand...)
The same way the real industrial revolution started: basic metalworking, leading to the steam engine, leading to machines able to perform tasks with more speed/power than people/horses. They would also allow generation of electricity which probably isn't much use immediately, but could become so quickly.
You don't even need excessive knowledge in the field: a basic understanding of how transformers, steam engines etc would be enough to get started. Educate others on these and they'll develop much of the rest themselves.