No, I wouldn't say that category theory is a prerequisite for domain theory in general.
I'm most familiar with domain theory from Winskel's "Formal Semantics of Programming Languages" (great book, cited as Win93 in the linked PDF), which has a whole chapter on the subject with no reference to category theory.
If you get deeper into domain theory (e.g. by working through this book) surely some category theory will be useful, but it's not a foundational prereq.
I'm most familiar with domain theory from Winskel's "Formal Semantics of Programming Languages" (great book, cited as Win93 in the linked PDF), which has a whole chapter on the subject with no reference to category theory.
If you get deeper into domain theory (e.g. by working through this book) surely some category theory will be useful, but it's not a foundational prereq.