How can he be both 22 and 42 when he became king?

The answer is given by Rashi on 2 Chronicles 22:2 (excerpt):

• (II Kings 8:25) states: “In the twelfth year of Joram the son of Ahab, etc., Ahaziah the son of Joram reigned,” and another verse states (ibid. 9:29): “And in the eleventh year of Joram son of Ahab, Ahaziah became king.” Perforce, his father did not die until the twelfth year of Joram son of Ahab, because it is written (ibid. 8: 16f.): “In the fifth year of Joram the son of Ahab, etc., Joram the son of Jehoshaphat became king, etc., and he reigned eight years.” And that which Scripture states further, “In the eleventh year of Joram,” is because he reigned one year during his father’s lifetime since his father was smitten with severe illnesses, as it is written (above 21:18.): “And after all this, the Lord plagued him, etc.” (22: 1) “And the inhabitants of Jerusalem made Ahaziah, his youngest son, king,” for one year during his father’s lifetime, and that is what is written: “In the eleventh year of Joram.” And where Scripture says: “In the twelfth year of Joram,” it means that after his father died, it was the twelfth year of Joram.

So what happened was, Joram became sick, so his son ruled as king, but could not be crowned king officially while his father still lived. The people of Jerusalem made Ahaziah king during his father's lifetime since he was smitten with severe illness. If this is not enough, please listen to Dr. Floyd Jones

Thus the sense of Ahaziah’s being “a son of 42 years” in his reigning is seen to refer to his being a son of the dynasty of Omri which was in its 42nd year. Putting the two Scriptures together reveals that Ahaziah was 22 years old when he began to reign during the 42nd year of the dynasty of Omri, of which he is also an integral part.3 [emphasis in original]–Dr. Floyd Jones,
“Chronology of the Old Testament”