For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Act I
Oddly enough, given how anti-chronological Catch-22 is, Act I really does take place at the beginning of the book. We may not find out the "Initial Situation" of the novel until somewhere near the middle of the narrative, but the opening material does lead us to a "point of no return" at the end of the first third of the book. That point begins in Chapter Twelve with the announcement of the mission to Bologna. It's this mission that leads Yossarian's commanders to realize just how deep Yossarian's insubordination runs. By redrawing the battle lines on the map and then turning back in the middle of the mission, we see that the only way Yossarian's conflict with the army is going to be resolved is through drastic intervention – though we don't know yet what that intervention might be.
Act II
The second third of the novel is given over to the accelerating decline of many characters' fortunes, most notably Kid Samson, McWatt, Chief White Halfoat, Orr, and even Doc Daneeka. As the absurdity of the novel becomes concentrated – think of Milo's multiple mayorships, Nately's weird Oedipal resentment of his prostitute's father figure, and Yossarian's own impersonation of the soldier who saw everything twice – Yossarian's impending rejection of the irrationality of the military complex seems inevitable.
The conclusion of Act II, when the novel seems as far from resolution as possible, comes in Chapters Thirty-Eight and Thirty Nine, when Nately's prostitute and her kid sister are kicked out of their home by the military police. Yossarian, AWOL with Milo and refusing to fly any more missions, is in Rome, where he was sent by the brass to recover from the death of Nately. Yossarian has become so thoroughly paranoid that he persists in walking backwards to prevent anyone from sneaking up on him.
As he begins searching for Nately's prostitute, and especially for her kid sister, he encounters a series of scenes of arbitrary and unstoppable violence; he also finds the ultimate definition of Catch-22: "Catch-22 says that they have the right to do anything we can't stop them from doing" (39.21). Yossarian has been unable to save either Nately's prostitute or her kid sister, both of whom come after him with knives; he has also reached a stalemate with the military, unable to leave but unwilling to stay.
Act III
The final act of this novel, and by far the shortest, runs chronologically from Chapter Forty, when Yossarian is offered Cathcart's dirty deal, to Chapter Forty-Two, when Yossarian flees for Sweden. The third act is the one when everything becomes clear and, despite all of the switchbacks and side plots of Catch-22, the final unraveling is structurally standard.
In the third act, all is explained: we finally hear Snowden's secret (see "Classic Plot Analysis – Denouement"); we realize that the novel has destroyed most of its characters; we see the final bureaucratic temptation of Yossarian; and then we observe our protagonist turning up trumps ethically, as he decides to run to Sweden rather than give up his resistance to the American military establishment. Yossarian's fate may be uncertain, but, in the end, his refusal to compromise his principles is really all the denouement we need.