Unfortunately all the evidence says otherwise.
Great managers can overcome the odds on occasion, which is what Klopp may do. But doing it consistently is rare. Nobody has ever really managed it over a sustained period of time. Wenger about the best example, but the last 10 years showed how that wasn't sustainable from a league challenging POV. And their budget was still pretty healthy tbf, at least from a top 4 perspective.
Liverpool will no doubt give a few of the teams above them a bloody nose from time to time, but without serious structural reform at the club (i.e. not just the manager), or a financial apocalypse at one of the Big 4, it'll be a one season up, one season down sort of thing.
He's a good coach, not a miracle worker. Dortmund were a solidly #2 team in Germany for majority of his reign from a financial perspective. He actually spent substantial amounts of money (12th highest spending coach of all time). So stretching them to finish #1 wasn't quite the same as taking a #5 and finishing #1, though obviously top 4 is very possible.
Really?
http://swissramble.blogspot.de/2012/10/borussia-dortmund-back-in-game.html
That's after their second league title.
tl;dr:
In fact, over the last three seasons no fewer than nine clubs in Germany’s top flight have spent more than Dortmund’s net €2 million.
Dortmund only really (net) spent big (~€60m) in his last season