From an earlier answer of mine plus check out /u/commiespaceinvader 's answer on how Muslim's were treated in the Third Reich.

It is interesting to note that the OP's quote came from Michael Mallmann and Martin Cüppers's book Halbmond und Hakenkreuz. Das Dritte Reich, die Araber und Palastina, and the original source, Nicosia's Nazi Germany and the Arab World disagrees with a broad swath of their conclusions. This is somewhat of a common attitude among specialists. One of the basic problems of Halbmond is that its source base for the Arab equation is sketchy to near non-existent. Götz Nordbruch's review in Welt des Islams noted that Arab voices only came from German-language sources and thus presented a highly distorted picture. Nicosia's review in Holocaust and Genocide Studies contended that Halbmond does little to contextualize both the Nazi's Arab policy but also antisemitism among various Arab leaders. Johannes Due Enstad 's review in the Journal of Contemporary History contains this interesting critique:

Neither Mallmann nor Cüppers knows Arabic, and so the documentary basis for their sometimes sweeping claims leaves something to be desired, consisting as it does of secondary literature and reports from German officials.

While Halbmond uncovered important details of Nazi Arab policy, including a Sonderkommando for a victorious Afrika Korps, it is not a very good book to understand Arab nationalist responses to Nazism and the war.

The influence of Nazism on various pan-Arab nationalist groups is highly contentious and politicized. There is no shortage of popular commentators post-9/11 that build on linkages between the Arab nationalists in the 1930s and early 1940s and the Third Reich. In some cases, this builds up to a charge that Arab nationalism was a species of fascism and shares many of its precepts of global domination. Conversely, there are commentators that deny the importance of Nazism to the political activism of the thirties and rejects any political connection between fascism and Arab nationalism. Neither approach presents an accurate picture of this period and its complexities.

In terms of form and some content, the Arab political movements of the 1930s did take some inspiration from the European political scene. Although the NSDAP and Germany were one model to emulate, it was Mussolini, not Hitler, that elicited more interest. With its corporatist model and more intellectually rigorous approach to politics, Italian fascism presented a much more relatable phenomena than Hitler's very German-centric brand of politics. Many Arab nationalist movements within the 1930s looked to Italian fascism as an organizational model, especially its paramilitary formations and leadership cult. The corporatist model had also seemingly worked in Turkey, which added to the attraction of the Italian paradigm in the MENA region. Arab political groups were far from alone in emulating the fascist model. Some Zionist groups and many European nationalist movements such as O'Duffy's Blue Shirts took cues from il Duce in the form of organized paramilitarism and a more militant form of nationalism. Fascism's promise of national regeneration and strength had currency in the context of the demi-colonialism of the Mandate period as it provided both an organizational model for resistance as well as a post-colonial model for government.

One charge often leveled against Arab political movements by the likes of Sam Harris was that they imbibed Nazi antisemitism and this has metastasized in the current invective against Israel. The Mufti of Jerusalem, Haj Amin al-Husseini, was an avowed Nazi sympathizer and active supporter of the Third Reich's actions against the Jews. He did conduct propaganda broadcasts during the war to stir up Palestinians against Zionists and the British occupation. Jeffrey Herf has made much of these broadcasts in his monograph Nazi Propaganda for the Arab World, in which he lays the origin point of Arab antisemitism in these wartime broadcasts. But Herf never really proves that this propaganda found a wide receptive audience within the Arab world. Very few Arabs owned shortwave radios and there is no real evidence one way or the other that they listened to the Third Reich's propaganda. Herf asserts that Sayyid Qutb might have been influenced by these broadcasts, but Qutb's antisemitic essays only appeared in 1951 after the first Arab-Israeli War. Joel Beinin's review of Nazi Propaganda for the Arab World sums up this problematic method rather well and is worth quoting at length:

Herf speaks ominously about “the merging of National Socialist with radical Islamist anti-Semitism and its diffusion to the Middle East” (p. 11). There is no easy way to determine how many Arabs listened to Nazi propaganda broadcasts or their reactions to the message. So, Herf resorts to speculative formulations like, “It is plausible that during World War II, Sayyid Qutb listened to Nazi broadcasts and traveled in the pro-Axis intellectual milieu of the radical Islamists in and around Al Azhar University” (p. 259). We don't know what radio broadcasts Qutb may have listened to. However, because Adnan Musallam's biography of Sayyid Qutb is in his bibliography, Herf should know that it was not until his visit to the United States from 1948 to 1950 that Qutb repudiated Western liberalism and began writing about Islamic themes (Adnan A. Musallam, From Secularism to Jihad: Sayyid Qutb and the Foundations of Radical Islamism [Westport, Conn.: Praeger Publishers, 2005]). Qutb did not join the Muslim Brothers until the early 1950s. His anti-Semitic essay “Maʿrakatuna maʿa al-Yahud” (Our Battle with the Jews) appeared in 1951, after the first Arab–Israeli War. This chronology, although it in no way excuses it, establishes a very different context for Qutb's anti-Semitism. The case for a link between Nazi propaganda and Sayyid Qutb's views about Jews rests on conjecture and innuendo, as does much of what Herf offers when he ventures beyond the terrain of Germany.

Far more persuasive than Herf is the work of Israel Gershoni and James Jankowski who assert that wartime fascist propaganda fell on deaf ears within most of the Arab world. Most Arab public intellectuals saw fascism as a more malignant form of Western colonialism and saw German propaganda as self-serving. Arab political groups may have taken cues from fascism in terms of political self-styling and aesthetics, but this did not translate into sympathy for the fascist cause during the war. Meir Litvak and Esther Webman have backed up this assertion by a deep analysis of Egyptian newspapers that reveals that between 1945-46, they regularly and accurately reported on the Holocaust and expressed a degree of sympathy with the Jewish victims. The Arab-Israeli wars led to a sea-change in Egyptian papers and then is when Holocaust denial and minimization gained traction within the Arab public sphere.

Even al-Husseini's extremism and involvement with the SS needs a proper contextualization. The Germans never devoted much in the terms of resources or attention to the Arab cause, and most of the conventional bureaucrats in the Italian and German foreign ministries favored other Arab leaders. After the failure of the Iraq revolt in 1941, al-Husseini ended in Europe and worked for the German propaganda machine. But in doing so, he had to compete with other anti-British Arab emigres and navigate the various factions within the German state apparatus. Al-Husseini's anti-Jewish sentiments put him in ideological resonance with the SS, but in light of other emigre competitors like Rashid Ali al-Gaylani, making an alliance with this particular power bloc had a certain political logic behind it. Al-Gaylani's more secular and pan-Arabist politics had found support both in the Italian and German foreign ministries and he was in many ways a more "current" leader against the British, having participated in the Iraqi revolt in 1941. Al-Husseini's main stand against the British had occurred a generation earlier in 1929. Making the most of his connections to Himmler was a way for al-Husseini to gain the paltry resources the Axis devoted to the Arab cause away from an emerging rival.

One of the problems with saying the Arabs "borrowed" elements from fascism is that it denies Arabs real agency in creating their own political movements. As the case of al-Husseini demonstrates, there was a political logic to the actions of some that cannot be simply ascribed to sympathy for fascism and its ideology. Conversely, denying the presence of any Arab engagement with European politics projects an insularity onto the Arab public sphere that borders on the farcical. Of course Arab public intellectuals and ideologues were aware of wider events in Europe and adjusted their movements accordingly. Writers like the Egyptian pan-Arabist Ahmad Hasan al-Zayyat could alternate between condemning Nazism's racial theories and extolling the virtues of Germany's economic regeneration under Hitler. There were elements of fascism that attracted Arabs in the interwar period, but also elements of its ideology many Arab nationalists found repellent.