The Spanish playmaker made the move from Real Madrid to the German champions in August and has quickly become crucial to Pep Guardiola's side.
Featuring in defence and midfield, Alonso has helped make up for the void left by injuries to the likes of Bastian Schweinsteiger, Thiago Alcantara and Javi Martinez, with Bayern leading the Bundesliga by four points after seven games.
Guardiola also brought in defender Mehdi Benatia from Roma during the close-season, and Bayern president Rummenigge has been impressed by both during the early stages of their careers in Bavaria.
"I knew the player Xabi Alonso well but the person off the pitch slightly less. From the first day I noticed that he has a great personality," Rummenigge told Goal.
"That's resulted in his style of playing football. He completed his debut at Schalke, having not even trained with the team before. Everyone has figured out that he strengthens Bayern and the transfer was extremely useful.
"I hope he will continue on this path. Benatia will give us additional quality in the next weeks and months - the stubborn negotiations have paid off."
Alonso and Benatia were just two of Guardiola's recruits in the recent transfer window, with the likes of Robert Lewandowski, Pepe Reina and Juan Bernat also arriving.
And having won the Bundesliga and the DFB-Pokal in his debut season last term, Rummenigge backed the Spaniard to deliver more success.
"Pep Guardiola is a very good trainer. He works with an intensity I have never experienced and I've been with this club for a long, long time now, he added.
"He’s an unbelievable trainer. We can all be fortunate he’s at Bayern. I notice that when I talk to players - he is a factor in why they want to come to Bayern.
"Robert Lewandowski and Mario Gotze moved to Bayern as they wanted to work with him, among other things. That's a plus for us.
"Sometimes he comes on non-training days, sits five, six hours in his office and analyses matches. On Sunday at Oktoberfest he stayed for two hours and then got on a plane to Italy to watch [UEFA Champions League opponents] Roma against Juventus.
"There are coaches who would rather drink the occasional beer more but he flew to Turin to watch our next opponent. He’s a meticulous worker."
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