This is a most interesting question, as there is actually something of a dearth of primary accounts for the latter emperor and a plethora of accounts (primary and secondary) for the former dictator. I shall attempt to answer both, though for further reading I highly recommend Christian Goeschel's Hitler and Mussolini: The Forging of the Fascist Alliance. Let's begin then with Il Duce.

The general view of the Italian dictator prior to the war was one of widespread admiration amongst the Nazis. Though keep in mind that I refer of course, to the upper leadership of the NSDAP, as to generalise and label Nazis to all Germans is a gross error. Hitler of course, was particularly interested in how Mussolini had managed to consolidate his power so swiftly and quickly after the Treaty of Versailles, whilst his own efforts in the 1920s were almost thwarted, first by the failure of the Munich Beer Hall Putsch and later by the "Golden Years" of foreign minister Gustav Stresemann. This admiration continued even as Hitler became Chancellor and later Fuhrer of Germany, so much so that in his first visits to the Italian state he remarked often of Mussolini's initiative and the bold moves of the Italian Fascist Party. As with many opinions in the Third Reich, this view disseminated into the upper ranks of the party as well, as Goebbels' propaganda machine churned out posters and propaganda portraying Hitler and Mussolini as (in the words of Goeschel)

"The Dictatorial Couple who would forge a new world order in Europe and beyond".

The reality however, was that after the war began Hitler became immediately skeptical of the Italian Army's capabilities and urged caution in their deployment to Mussolini. After the embarrassment of the Greek and North African campaigns in 1940-1941, Hitler's opinion of his Italian counterpart began to deteriorate significantly (as did the opinion shared by many of his fellow Nazis). By about 1943, the wartime meetings that the two were having were more so dedicated to "propping up their image" rather than actually deciding on any tactical or strategic joint operations. The Teutonic leadership of the Germans began to increasingly view their Italian allies as more of a burden rather than a benefit to the Axis Alliance; so it came as no surprise in 1943 that the Allied invasion of Sicily also aimed to exploit the weaknesses of this relationship by threatening the weaker of the two states. By 1944, Mussolini was little more than a puppet for Hitler, and his assassination elicited no major remarks of grief from Berlin.

As for Hirohito, distance more than actual problems prevented the Nazis from forming their own comprehensive view of the Japanese Emperor. However, from secondary sources such as this WarfareHistoryNetwork article, we are made aware of the fact that the Nazis certainly viewed Hirohito as an equal to them if not ever so slightly inferior (though that was because of his Asian background rather than his capabilities). The Nazis definitely did hold ceremonies in celebration of the alliance with Hirohito's Japan, but beneath that facade we are still uncertain as to whether the general populace had any wildly differing opinions on this foreign monarch who wasn't technically a Fascist; though the ultranationalist, imperialist nation which he led was not exactly looked down upon either.

You may have heard that Hitler conferred upon the Japanese the status of Honorary Aryans (Ehrenarier), an indication of not only the solidarity between the two nations, but also something of a reflection of Hitler's own beliefs. Take a look at this short extract from The Testament of Adolf Hitler, literally a part of his final documents and later his will:

Pride in one's own race – and that does not imply contempt for other races – is also a normal and healthy sentiment. I have never regarded the Chinese or the Japanese as being inferior to ourselves. They belong to ancient civilizations, and I admit freely that their past history is superior to our own. They have the right to be proud of their past, just as we have the right to be proud of the civilization to which we belong. Indeed, I believe the more steadfast the Chinese and the Japanese remain in their pride of race, the easier I shall find it to get on with them.

If anything, the quote certainly shows that Hitler viewed the Japanese as worthy of a place in the new world order his Nazis were attempting to create.

So there we have it, two vastly different Allies, two vastly different courses of opinions. Whereas Mussolini's Italy was held initially in admiration but later wartime contempt, Hirohito's Japan was viewed with steadfast appraisal and historical admiration throughout the course of the conflict. Hope this helps with your question!