There are a couple good answers already about how Paul can help the world prepare by sharing knowledge, or advancing technology. But if he is really prepared to consider any option, to take any action, there are at least a couple more options he should consider, whether or not he actually decides on using them.
Take up Super-villainy
Honestly, there is very little that pressurizes R&D and advances technology like a real, genuine, threat. While it is possible that he can take advantage of existing tensions to market his new & improved defense technology, it isn't guaranteed - especially if the discrepancy between the incoming aliens and Earth is significantly greater than the discrepancy between what a nation can do with what Paul-advanced tech is capable of, and what the next most powerful nation can come up with in the intervening time. It is quite possible - if one nation gets too far ahead, or if (kinda like nukes) the tech becomes sufficiently dangerous that no one wants to use it on their own planet/people - it can stall out with either that one nation suppressing others and not needing more or better weapons, or with a Mutually Assured Destruction pact, which prevents the tech from being used/developed more - potentially before it reaches the levels needed to defend against the aliens.
So, Paul might consider becoming a super-villain. As an individual, he can become a threat, to his specified level, and and spark innovation that way. Better yet, if he plans well enough, he can unite the best minds into working together against him, instead of against each other - and he might make strategies that develop piecemeal the tech he wants (better targeting systems to foil this scheme, now tighter focus to fix that, and preventing this kind of target jamming, how to avoid overloads) which can happen to be pieced together into the super-effective weapons and systems needed when the aliens come (as targeted as he can make it), instead of being halted as not-needed, if it was not-needed against purely human threats.
He might not decide that is the right path to take - if nothing else, it will make his life miserable, paint a target his family and hometown, and if he fails at all (is caught, is out-developed, is killed) he will fail totally since no one will believe him after he makes himself that level of threat (or nuisance). But if he's honestly willing to look at all the possibilities, he should at least plan it out and see how it stacks up against other plans, to see which offers the best chances. Of course, since part of this idea's problem is how it will ruin his own life and credibility, he might try a related solution instead.
Make a Super-villain (hoax)
Hey, if he doesn't want the job, why not give it to someone else - preferably someone entirely imaginary, since then they won't get out of control. Probably. Hopefully.
Anyway, he will need to plan some kind of background story for this villain, including a pretend-goal, and some kinda logic for how his actions will serve that goal. He will need some remote disguise, so it isn't clear he's imaginary - maybe a robot, or a costume, some signature effect, a signed note, whatever works - so people will know who this is supposed to be. Also, he will need to pre-develop some fairly advanced tech (again, as close to the alien tactics as possible) so he can have a "head start" on the rest of the world, and be a viable enough threat to the rest of the world to get the weapons up to par.
As a bonus, this way he can be on the side of angels as long as the hoax doesn't get revealed - collaborating with them, influencing the direction of their inventions (and using that knowledge to keep the Villain's threat relevant). And, if he develops counters and counter-counters at the same time, he can get into a technology race with himself, and "release" technology upgrades targeted towards fighting the aliens from both sides, with a bonus of whatever the other human weapons developers come up with to make it better.
It is possible, if he manages the threat level vs actual damage well enough, and has a good enough rapport with people he has been collaborating with - he might be (kinda) listened to even if the hoax is discovered. If they believe he at least believes in that level of threat, if he has made the villain frightening but not directly caused that much damage (via "last minute saves" by himself or various deus-ex-machina)... they might be willing to develop the tech just in case, even if the hoax costs him their trust otherwise, and possibly his freedom and/or future. And wouldn't it be worth it, for the whole world?
Make an Alien (hoax)
Well, maybe he doesn't want to cause that much damage, even if it does get everyone working together and the right sorts of directions to nullify the threat. But, having already had the idea of a hoax - can he try a different one? Make up some "alien refugees" (or possibly just the one) - maybe they can even be "advanced warning" of the alien threat, having barely survived the same species' advances (and possibly "lost their own planet" in the bargain).
This will be similar to the "super-villain hoax", since Paul will want to avoid any scientific experimentation (including simple bio-analysis) that can prove it's a hoax. On the other hand, any actual alien refugee would also want to avoid being experimented on, especially if they saw some of our storybooks and alien treatment in them, first. So, having the refugee(s) openly state their "reasons" for hiding will help. Gifts of advanced technology (developed by Paul, of course) can be given - preferably to multiple governments simultaneously, and with carefully leaked media exposure - along with the warning of the alien fleet, its estimated arrival timeline, and the potential fate of earth, based on the "fate" of the hoax-alien planet, if the alien fleet isn't resisted.
Obviously, it's best if the technology was a boost in the right direction. Possibly the information is "fragmented" or partial, openly attributed to the hoax-aliens laws about advanced tech or limited by what they "could scavenge", or what "they" (Paul) thinks is safe to give humans, and which actually depends on what Paul can actually come up with and what time-frame he is willing to give to his own tech development vs the world's response time. Human scientists can go over the data and develop ways to fill the gaps on their own.
He may want to stack the deck of his hoax with media exposure, his tech or robots or whatever proxies publicly offering help in disaster scenarios, using their "advanced tech" to give warnings or open aid, kinda working the superhero trope and giving concern about human reaction as a reason for not coming forward openly. As long as everything is done through proxies, it will be hard to prove it's a hoax - especially if the tech involved can really be developed by Paul. And this way it will be possible to openly warn the world about the alien threat - nobody will be surprised or doubt that space aliens believe in and/or know about a space alien fleet!
Again, it is possible that if even if the hoax is discovered, he might be able to convince the discoverers that he has a genuine belief in the coming fleet - he has proven his intelligence with developing all that tech and the hoax itself, yeah? And at least in this hoax, Paul and his "refugees" have been helpful and not caused damage, not acted as a threat, gifted technology and weaponry (to multiple nations, no less). So they might continue developing the technology just in case it's true - or even let the hoax continue, with people not in the know innovating in the meantime, until the invasion happens (or doesn't), thinking they can decide what information to release then.
Start a religion (hoax)
Actually, if he's already figuring out what to do to make villain or alien hoaxes work, why not check into what it would take to make a religious and/or cult hoax? As a bonus, he can even do this openly after a reasonable set-up period. He needs s good understanding of religion, psychology, and cults - both sincere and scams - to make it work, but he's intelligent enough to figure something out.
So, Paul has been blessed. Really, he has been - and he can prove it, with all his shiny gifts of physical, mental, and educational enhancements! Let him gain some money, build some influence, look at dominant religions and what suits his needs, and see if he can't build a cult out of it. He can (as in other approaches), develop advanced tech to make the cult dangerous, powerful, able to meddle according to their creed. He might not want to start with alien-fleet threats (unless he's really sure he can get enough believers) - but powerful, and armed, and willing to develop some dangerous technology will get attention.
Depending on the substance of his cult-message, some of that advanced tech can be used to "make miracles" via robots, remote tech, all kinds of things. Otherwise, the ability to develop so easily can be, er, given the question maybe that should be "actually is" a gifted miracle in its own right - if and when he has enough people, enough influence, enough trust, to undergo whatever testing to prove his various blessings. He can, again, develop and give away technology to various nations or interests (in accordance to his stated creed). He can give warnings about a coming menace - especially once he's bought influence with his tech and has some followers, he might actually be listened to. He can inspire generous or beneficial actions in his followers, to gain influence - or ambiguous or dangerous actions, to create a threat to work against.
Paul has to start small, and build a network of followers, a cache of technology, and some serious influence before he can come out and talk about the alien threat - but unlike some other methods, he can talk about it. He can claim his enhancements openly, give the real backstory - with a study of psychology and history, he can figure out how to persuade people quite well. And, he can use advanced tech to "fill in the gaps" of his cult's beliefs, creating miracles, goodwill, or "supernatural" threats to get people prepared for the alien invasion.
or Just buy the technology!
Paul, with his intelligence and insta-knowledge, can probably come up with some ways to get rich, fast. And between wealth and intelligence, he's going to get influence. Have him offer (conditional) grants in scientific fields he believes useful to R&D, have him invest in weapons companies, communications and jamming, anything that might help when the day-of comes. He can get pretty far capitalizing on just the human-level threats, between national interests - and get a good tech-boost out of it, especially if he invests himself and his intellect along with the money.
The difference here is, as he is gaining influence and attention - he can quietly take carefully chosen people into his confidence about the reason why. Wealth and influence will shield him from a lot, obvious intelligence will shield him from some more of the consequences if someone doesn't believe him - not all of them, not by a long-shot, but enough he doesn't have to worry about being dragged off to the loony bin, and he can let people sniggering at him go. People buying the weapons systems won't really care if they're getting lasers instead of missiles because their backer believes in aliens, they will care that the lasers are powerful and precise and destructive enough to replace the missiles as the go-to weapon.
And while he might be able to convince people along the way, enough to avoid leveling off of tech or de-prioritizing weapons R&D because they think the weapons they have are good enough - even if he can't convince them or doesn't try, he can plow cash into research piecemeal that happens to work together - this bit of software helps the target systems, that has to do with increasing power, that little thing helps efficiency - so that when it's time, the pieces just need to be assembled to get your weapons and systems designed to be effective against the alien tech.
This is pretty much the same work Paul was doing in the last couple options - what tech, and what tweaks, will steer people in a direction effective against the alien fleet, and what human tech can be developed around or between these requirements. In this case, he just uses grants to buy the individual research pieces originally (which might not even look like anything dangerous till assembly-time), and only pitches the weaponization of the systems and parts once the background, theoretical research is done.
Again, some of these possibilities are more pleasant than others. But if Paul is serious about the threat and his need to get people ready, he should at least consider them - even if he ends up not using them in favor of a different plan.