Were the Nazis actually ever horrified of anything that the Imperial Japanese Army did or vice-versa?

Yes, at least one member of the Nazi party is known to have been horrified by the actions of the Japanese Imperial Army.

It depends on what you understand by the (plural) term

the Nazis

I would, in this historical context, understand this as the

• Government or leadership of Germany or Japan

both of which, together with Italy, were allies with a common goal.

With one major exception

• Japans unwillingness to concentrate on the defeat of the Soviet Union in the east

they did not care how the other achieved that goal.

That one member of the Nazi party, John Rabe was horrified by the events is clear, based on his diary entry from December 13, 1937:

Last night up to 1,000 women and girls are said to have been raped, about 100 girls at Ginling College ... alone. You hear nothing but rape. If husbands or brothers intervene, they're shot. What you hear and see on all sides is the brutality and be*t**lity of the Japanese soldiers.

and his public actions:

Rabe's attempts to appeal to the Japanese by using his Nazi Party membership credentials only delayed them; but that delay allowed hundreds of thousands of refugees to escape.

Even after his return to Germany (February 28, 1938) these efforts continued:

Rabe showed films and photographs of Japanese atrocities in lecture presentations in Berlin and wrote to Hitler to use his influence to persuade the Japanese to stop any further inhumane violence. As a result, Rabe was detained and interrogated by the Gestapo and his letter was never delivered to Hitler. Due to the intervention of Siemens AG, Rabe was released. He was allowed to keep evidence of the massacre, excluding the film, but was not allowed to lecture again or write on the subject.

This I think this is a telling sign, that although exceptions existed, as a general rule the Nazis did not care what their Japanese allies did.

After the war, Rabe was arrested first by the Soviet NKVD and then by the British Army. Both, however, let him go after intense interrogation. He worked sporadically for Siemens, earning very little. He was later denounced for his Nazi Party membership by an acquaintance. He was stripped of the work permit that he had previously been given by the British Zone, and had to undergo a very lengthy de-nazification process (his first attempt was rejected and he had to appeal) in the hope of regaining the permission to work. He had to pay his own legal defense costs, which depleted his savings.

Unable to work to support his family and with the savings spent the family survived in a one-room apartment by selling his Chinese art collection, but this did not provide enough to avoid malnutrition. He was formally declared "de-Nazified" by the British on June 3, 1946 but thereafter continued to live in poverty. The family lived on wild seeds that the children would eat with soup and on dry bread until that was no longer available either.

In 1948, the citizens of Nanking learned of the very dire situation of the Rabe family in occupied Germany and they quickly raised a very large sum of money, equivalent to US$ 2 000 ($ 21,000 in 2019). The city mayor himself went to Germany, via Switzerland where he bought a large amount of food for the Rabe family. From mid-1948 until the communist takeover the people of Nanking also sent a food package each month, for which Rabe in many letters expressed deep gratitude.

The fact that the citizens of Nanking remembered and assisted John Rabe after the war, is a sign that his activities at the time were generally known there.

The quoted source Rape of Nanjing - RationalWiki, is in my opinion, too one sided and mainly based on a single source:

• The Rape of Nanking: The Forgotten Holocaust of World War II by Iris Chang

The carnage was so bad that the local Nazi leaders decided to intervene and put a stop to it.

• this statement is, I believe, a conformation of the claim that Iris Chang book is, in areas, seriously flawed
  • one member of the Nazi party is not the local Nazi leaders
  • and that they: put a stop to it, as the statements claims, is not true

So a critical look at this one statement leads me to conclude: that this a piece of misinformation (whether through ignorance or deliberate).

The corresponding Nanjing Massacre - Wikipedia statement gives a more realistic description of the situation:

A group of foreign expatriates headed by Rabe had formed a 15-man International Committee on November 22 and mapped out the Nanjing Safety Zone in order to safeguard civilians in the city.
...
Rabe and American missionary Lewis S. C. Smythe, secretary of the International Committee and a professor of sociology at the University of Nanking, recorded the actions of the Japanese troops and filed complaints with the Japanese embassy.
...

which describes a completely different situation and uses multiple and more detailed sources:

By February 5, 1938, the International Committee had forwarded to the Japanese embassy a total of 450 cases of murder, rape, and general disorder by Japanese soldiers that had been reported after the American, British and German diplomats had returned to their embassies.

Woods, John E. (1998). The Good Man of Nanking, the Diaries of John Rabe. pp. 275–78.

Some of the academic criticism quoted about Iris Chang book states:

Joshua A. Fogel, at York University, argued that the book is "seriously flawed" and "full of misinformation and harebrained explanations." He suggested that the book "starts to fall apart" when Chang tries to explain why the massacre took place, as she repeatedly comments on "the Japanese psyche", which she sees as "the historical product of centuries of conditioning that all boil down to mass murder" even though in the introduction, she wrote that she would offer no "commentary on the Japanese character or the genetic makeup of a people who could commit such acts". Fogel asserted that part of the problem was Chang's "lack of training as a historian" and another part was "the book's dual aim as passionate polemic and dispassionate history".

Conclusions:

When you are building a 5 story house

• the previous stories should be built in a manner that they can support the stories on top of them
  • otherwise the house will collapse

The RationalWiki article as a source for this question

• is not a stable foundation for making a claim

since it is based mainly on a single source that is seriously flawed.

The Wikipedia article is a better source for this question

• since it is based on multiple sources that, independent of each other, can possibly confirm the correctness of the given statements

and therefore the likelihood is higher that the information contained in it is correct.

Wikipedia is good as a starting point for something that one is interested in, since it often contains the sources (or pointers to a source) that one can dig into to gain a better understanding.

However, as with all sources, they must be criticaly looked at (or compared with each other), but never be taken as an absolute truth (independent from which direction that truth claims to comes from).