I would say generally it is not how it works, even if there are examples of such history-changing technologies - bronze, iron, steam, metric system(standardization concept in general), transistors, lasers, personal computers. There are technologies which deeply affected consequent events. So as there are plenty of examples of stuff which was discovered randomly, so it possible to speed up certain things as part of the main goal accelerate technologies or getting the upper hand in competition to obtain resources and be able to focus them on some goal, in a similar way mentioned in Nosajimiki answer.
Generally, technology and the ability to do a thing is interconnected, it is a result of being able to do many different things, so as ideas have to come in the right time at the right tech level. An example handwriting input Apple Newton good idea but software and hardware was not up to the task. Same as today deep learning is a significant topic, but it would not do that much in the 90's as hardware was not up to the task, and advances on hardware tied to many other advances and specifics of technologies. And even if one would know and be able to remember all those aspects(and it really connected not only with the process themselves, but with other processes which produce production equipment you need for your process, and those other processes also may require advances in other production equipment ....
Is there or isn't there a way to achieve the goal of the character depends on the place where he is in the timeline. Is there a chance for him in the stone age to progress to the first PC in 100 years - I would say rather no, one of the big reasons he can't do everything required all by himself, so even if he has perfect knowledge he needs to teach others to do certain stuff he needs to do, and it creates a lot of problems.
Can we imagine some knowledge body carried by one human which allows him to jump from stone age to space exploration immortality - I would say yes, we could imagine such knowledge - but the problem of the guy is how far he is initially from it but it is not the only problem in the case. It can be imagined like a road, with some boulders/rocks on it (which may represent some key achievements, a bit more complex than that - achievements and the plan to reach it and next rock in a certain time frame from certain starting position(which is a constant in our case, which makes it harder)) and a condition that you need to jump from one rock to another to move to the goal - some gaps may be too big for one jump(it is the time frame of the guys existance). So, in general, it is the question, is there a path on the road(or better to say a field of rocks) - such so that one can jump from one rock to another and after a certain finite number of jumps reach the goal place.
It some sort of traveling not a fully interconnected graph, which structure is defined by starting conditions and by some virtual technology tree(sort of The Library of Babel of technologies). And even if we can imagine a technology which may be implemented with a small workforce(number of people), it may be like an island in the possible routes, an island to which there is no path, even if we can see it we not necessarily can reach it in given conditions.
Stone age is easy to demonstrate the complexity of jump, but at the same time if we look at it from our current time, biological components/technologies may be a way to reach the goal - microbiology may produce a lot of chemicals, and mutations are a potential tool so as a selection of strains of bacterias/cells/whatever may give you materials which one typically associate with our current time like plastic strains of bacteria that are able to produce plastic:
- Scientists at Genomatica Inc. recently announced that they have developed strains of bacteria that are able to produce plastic without the use of oil or natural gas. The sustainable process utilizes little more than sugar and water to produce butanediol (BDO), which can be manufactured into everything from plastics and fibers to pharmaceuticals.
Or another one, maybe it harder to track any connection with the topic here, but it is interesting in terms of it can be another chemistry tool, April 2017 scientists discovered that octopuses, along with some squid and cuttlefish species, routinely edit their RNA (ribonucleic acid) sequences to adapt to their environment.:
For a guy living events over and over and being able to experiment with the environment and those events, it isn't that impossible to come to an idea of small creatures living unobserved, and observing them took efforts of certain individuals which technologically were simple enough. Genetics is quite a jump, so as chemistry and the whole idea that those microorganisms can be used for some useful goals. But it is a powerful tool once one realizes it, with interesting potential, and which can produce a useful combination even if you do not know about its existence before. it just to show that stone age guy, in theory, has a chance too, but it very unlikely and to meaningfully propel to idea of plastic it needs to know the past of our time which is out of reach of the guy.
As for 2000's it is a good time for your goal to be. Distances between individual rocks are smaller(in some sense), but it more of those and they have to come in the right sequence and it really a lot of them (deep recurrence, much deeper than it was for old days) and there a lot of information to carry in memory to make next iterations possible. But it is helpful that there plenty of people and there is a well-established system which helps to focus efforts in a certain direction, so as relatively developed information processing, which may be helpful to carry information back as an example. As an example - there probably is some algorithm of pseudo-random number generator(idk couped with some markov chain type stuff), which is short enough and which has relatively short seed to remember and which produces a replica of encyclopedia of Britannica(in random order, or maybe even consequent(but it less probable)), or a description of how to build a warp drive, or something more useful at current interation of the character.
2000's are good as it was a time of many opportunities created by a rapid development of technologies in a short time and at a good pace and I would say to the guy - well done well done
What kind of technologies it could be as of today
- as story plot it is your choice of course, but I would say that most of the job is done by the time
In terms of achieving goals today is a good day for that, it possible to argue that we have everything we need for our next big jump in time and technologies.
SpaceX started as a mariachi band in 2002, now for quite some time, it delivers payloads in space and ready to deliver humans in space. And it has, theoretically, the capacity to deliver payloads on the moon.
With the moon situation, we have an interesting direction of Affordable, Rapid Bootstrapping of the Space Industry and Solar System Civilization which can be coupled with Realistic cost and feasibility of sun-shading? as mean to face global warming which may lead to implementing solutions like that ServerSky (and not necessarily just stop at 2%, go K1) Whis wich generative algorithm and software and general digitizing of researching for materials and solution may lead to burst in our knowledge of technological and biological nature using those space resources and computations.
Which if it does not solve the problem by itself, in the required time frame, which I doubt, may lead to sophisticated technologies like smart matter there some bits on that regard mostly as potential scale of the possibility provided by smart matter or it can be any gray goo of a flavor you like.
And gray goo type thing is probably suitable to be produced by means of tools we have today, but we are just lack of knowhows in that regard. And that allows you to go to K2 with from the same position as it was K1 here in the same period of time or shorter and it's a maximum power in terms of applying generative algorithms to the problems which the character is worried about and are maximum probabilities in solving them in the given time frame. (and I would say it easily can be done if the character covers only 2000-2100, so as I would argue that he could do that in 2000-2060)
Quantum computing seems another technology which we sort of close and able to do but lack knowledge of how exactly. Considering all those metamaterial discoveries which potentially can be used in theory for those etc - so it may be some right combination of specific knowledge. And QC it can be quite powerful to produce new knowledge (generative designs and all that again)