Shrek is a big mean ogre who wants to get rid of the fairy tale creatures dumped in his swamp and in order to get rid of them he accepts a deal that he first has to rescue a princess.

While I agree he definitely starts out as an antihero I'm not certain I agree that it's the rough, uncouth or ugly characteristics that makes him one but rather what drives him ("I want my peaceful swamp back!" isn't exactly an altruistic quest). But then, the definition of what a hero is isn't exactly clear either which makes defining the antihero all the more difficult.

Scholars have not yet reached a consensus about how heroes should be defined, which hinders theory development, hypothesis- testing, and communication with various target audiences (e.g., Rozin, 2009).
In recent psychological literature, heroes have been described as persons who: are willing to risk and to make sacrifices for others (Becker & Eagly, 2004); resist external pressures of conformity (Zimbardo, 2007); protect and promote the well-being of future generations (McAdams, 2008); persist in the face of failure (Ko, 2007); demonstrate moral will, the desire to do good for others, and moral skill, the capacity to do the right thing in a particular situation (Schwartz, 2009).
These varied definitions evidence a lack of consistency in conceptualizing heroes within psychology. We argue that these definitions, at least at the individual level, lack sufficient rigor and coverage (Gregg, Hart, Sedikides, & Kumashiro, 2008).
[...]
To our knowledge, three studies adopted a free-response format to study conceptions of heroes.
In 1997, Gash and Conway asked 700 children in Ireland and the United States to identify the features of heroes. Participants named their hero and described their hero’s features. The 24 features were: active, beautiful, brilliant, brainy, brave, caring, confident, dresses well, famous, friendly, funny, good, gentle, good-looking, helpful, honest, important, kind, loving, loyal, rich, skillful, strong, and warrior.
Further, Sullivan and Venter (2010) asked American college students to identify one hero and to provide descriptive words to explain why. Participants were provided with an example: “Hero: George Washington, U.S. President; Reasons: Honest, intelligent, great leader, brave” (p. 475). The features that emerged were: intelligent, loving, religious, caring, leader, talented, hardworking, motivated, role-model, and creative. Demand features may be reflected in the results due to use of an example prior to the free-responses.
Allison and Goethals (2011) asked American college students to list the traits that they use to describe heroes. Then, another group of students sorted the traits and revealed eight trait clusters: smart, strong, selfless, caring, charismatic, resilient, reliable, and inspiring.

• Zeroing in on Heroes: A Prototype Analysis of Hero Features - page 115
  Authors: Elaine L. Kinsella, Timothy D. Ritchie, and Eric R. Igou
  University of Limerick
  https://pubmed.ncbi.nlm.nih.gov/25603370/
  (Available on Annas Archive for those curious to read more)