It sounds like you are asking whether Updike uses both of these literary techniques, as opposed to one or the other. The truth is that the "Rabbit" novels, beginning with Rabbit, Run, all use some stream of consciousness and interior monologue at various times. This is not only true because each novel is focused on the eponymous character named Rabbit (Harry Angstrom); in fact, the novels also use those techniques for other characters.
The novel's style does not tend to differentiate the moments with quotation marks or italics for the most part, as the specific "speaker" is usually clearly identified because that chapter or segment is focused on their thoughts and actions. Reviewer Helena Cuss says the novel's style is "part stream-of-consciousness, with all the immediacy of the present tense."
Harry's wife Janice and his son Nelson in particular are portrayed in this way, as are Harry's thoughts and feelings about them. For example, in Rabbit, Redux, Nelson, Harry's teenage son, wonders aloud "Why doesn't Dad just die?" and it is not within quotes or even italicized but simply part of the narrative taking place.