The ideal method is to cheat. You can pick your battles, take on tasks where you have secret information or advantages, come prepared with more prep than anyone expected you to have done, and end up with a task that may well be very difficult to you and none other. In this as most cases, dath ilan considers cheating to be technique, and if it's sometimes a bit less impressive it makes up for it in ease and effort, but betting on a market that gives incredibly long odds against something happening it and then resolving it by their own actions is one of the archetypical impressive ways for a member of civilization to get rich.
There's probably a number of ways to do that here. The events of the past day have shown that many things they thought they knew about the state of the universe, and possibly the laws of physics that govern it, are wildly wrong, and where your map doesn't match the territory there are usually routes between points you cannot see. Many people are devoting their time to figuring those out, and if they can figure out how to make their own portals or whatever the trick is to this kind of bioengineering or whatever dealt with those helicopters, there's plausibly something applicable here. The problem here is that all of those are already being done by some of the brightest minds in civilization with enormous amounts of resources behind them, and coming in from a relatively cold start it's very unlikely that they'll be able to both pass them by and get it done fast enough to matter. The linguists have a rather simpler problem, in that there's is one that has been extensively prepared for with tens of millions of hours of labor in advance, but whatever simple structures underlay the emergent complexity of alien communication have proven difficult to derive and their example corpus leaves a lot to be desired. Absent a low probability development there, this method won't do them much good in the timeframe in question.
What, then, to do?
It's a well understood phenomenon in dath ilan that, once something has been accomplished once,accomplishing it again is typically simpler. Not just because they were able to learn from the previous attempts, either - knowing that a problem can be solved, and under what constraints this was done, is itself large amounts of data about what the solution has to look like. If you learn it is possible to build something in 10 hours, you can discard offhand all the search space with longer duration; from these and other details, it is often possible to figure out in broad strokes what the solution must have looked like for it to have been solved. If attempting to solve a problem normally involves first determining what you have, and then what you want, and how to use the former to achieve the latter, this involves looking at what success will look like, and determining what must have happened to achieve it. If something possible seems impossible, then that is not a contradiction, but a sign that somewhere important your map does not match the underlying territory. You need merely assume that a way to win exists, and if necessary reframe success and the material conditions until one pairing matches a way forwards. It can be a dangerous epistemic state to try and stay in for too long, even for a trained inhabitant of civilization. Sometimes there isn't anything wrong with your map, and there isn't actually any limit to how much you can throw away in pursuit of such a strategy if you're wrong enough about how the world works. And it's far from the only method, with dozens of other commonly used reframing schemas existing. It's just good enough to often see use regardless, and this is one of those times.