What does it take for a lower-ranked university to compete with a top-ranked one?

Money can indeed buy a lot of things that will move you up in rankings, though probably not all. It's true that you will be able to attract good faculty with money -- spent on salaries, research infrastructure, beautiful offices and spacious labs well equipped with machinery, several postdocs attached to each professor position, and maybe money to support (grad) students and travel. For this, you will actually need a lot of money -- take a look at the budgets of the really good universities in the US and UK, and you will realize that you will need an operating budget in the $1+B region per year. So $10B will be spent quite quickly.

But that by itself will not be enough. Faculty will generally only go to places where there are also good students, and the best students go where they can historically expect to get an excellent education. To find out where these places are, they look at, say, the top 20 or 30 universities in the world -- so you're caught in a catch-22 if you want to move up. In other words, you have to have a long-term strategy to move up: things can't be achieved within a few years, but it will take 10, 20, 30 years of big spending if you really want to get high in that list.

There are some examples of relatively new universities who have really tried this, and you can look at the press coverage to see what they have achieved and what they haven't:

• The King Abdullah University of Science and Technology (KAUST) in Saudi Arabia
• The Okinawa Institute of Science and Technology (OIST) in Japan
• National University of Singapore

There are other examples in China and in the gulf countries of universities that have (tried to) significantly rise in rankings in recent decades.

Suppose I'm the rector of the University of Bonn and I want to make my university as prestigious as Oxford. What will I need?

if I can find $10+ billion and several years of time, I will be able to make my university one of the world's best?

Your basic question is quite an interesting one, but the way you formulated it involves several false premises:

First, that there is a chance in hell for the rector of the University of Bonn to suddenly get his/her hands on $10B; there isn't.

Second, that if he/she were to "find" such a sum of money after all, that trying to buy the university's way up the Times of Higher Education rankings to overtake Oxford as the top-ranked institution would be a good way to spend the money; it wouldn't, and no sensible rector would try to do that.

Third, that the way to climb up the rankings would be as simple as buying good researchers, who "can be bought". There may be some kernel of truth to that assertion, but it isn't nearly as simple as you make it out to be.

All of those premises are false, and for somewhat interrelated reasons: the rector wouldn't get $10B (presumably from some rich donors who really like the city of Bonn) to try to climb up the rankings, because climbing up the rankings simply isn't a good goal to strive for (he/she also wouldn't get $10B for any other purpose, because this is too much money for a single university to spend efficiently). Think of it this way: the University of Bonn is already a perfectly good and functional institution that serves the German people and the world at large quite well as it is -- that is why it holds the already very respectable position of the 100th ranked institution on a widely-known global ranking of universities. What exactly would be the point for it to become #1 by poaching all the good researchers from Oxford and other top universities? Even if it were possible to do so (and it isn't really; see below), this would add nothing of value to the world. Arguably it would benefit the city of Bonn at the expense of various other cities, but what investor or philanthropist would spend $10B just to transfer a chunk of intellectual and economic wealth from one rich country to another? One can achieve much greater and more worthwhile things with such an amount of money.

As for buying good researchers, the reason it's not as simple at that is that people's decisions of where to build their careers are a lot more complicated than just how much they get paid. Top researchers want access to other top researchers, to top research infrastructure, to a nice city to live in where they like the food and the culture and speak the language, to top students (who in turn want some of the same things, and would also need to be "bought"), and much more. Oxford has many ingrained advantages over Bonn that would be very hard or even impossible to replicate with any amount of money - most importantly its presence in an English speaking country (compare English's position as the language with the largest number of speakers, estimated at 1.12 billion, compared to the German language's status as the 12th most spoken language with 132 million speakers [source]). So yes, a lot of money could certainly buy you a few top researchers here and there who are willing to relocate late in their careers in pursuit of a salary even higher than their already-high one; and hiring such researchers is generally speaking a pretty good approach to getting started on the path to improving your university's quality (not "prestige", which is absolutely the wrong thing to strive for; quality should come first, and prestige is what you'll get once you have high quality). But that path is a much longer and more tortuous one than what your question suggests. A hundred billion dollars and one hundred years, as well as wise stewardship of those resources, might indeed propel an institution to a very high place in the rankings (think of places like MIT or Caltech), but $10B and "several years"? Sorry, no chance.

As a way of summarizing this long answer, I'll respond to your question if it's the case that Oxford is more prestigious than the University of Bonn because it is richer. No, that's not at all the case. Rather, it is more prestigious because it is better, and it is richer also because it is better; and it is better mainly because it is much older (and of course, because its leaders spent the time since its founding quite wisely rather than squandering it away). The money does enable it to stay better, but another university cannot simply replicate all of Oxford's advantages just by throwing money at the problem for a few years.

Find hundreds of researchers that do not care about rankings. Give them freedom, don't make any pressure to force them to play any ranking-related numbers game such as papers or third-party grants. If you are lucky, some of them may produce great results in 30 or 50 years. Then your university is the best one in the world.