If his wife becomes a Muslim then apostatizes, what should he do?

Guidance to the religion of Islam is a great blessing, and
the one who is the cause of a person being guided will have a great reward.
It was narrated from Sahl ibn Sa’d (may Allaah be pleased with him) that he
heard the Prophet (peace and blessings of Allaah be upon him) say to
‘Ali ibn Abi Taalib: “By Allaah, if one man were to be guided at your hands,
it would be better for you than red camels [i.e., the most valuable kind].”
(Narrated by al-Bukhaari, 2942 and Muslim, 2404). 

So you must do your utmost to guide your wife to Islam
because of the great good there is in that for you, for her and for your
children. 

With regard to her situation after becoming Muslim, this is a
matter that rests with Allaah, because you do not know what will happen to
you, will you remain steadfast in Islam or not? For people’s hearts are
between two of the fingers of the Most Merciful and He turns them as He
wills, so how about someone else? It was narrated from Anas ibn Maalik that
the Messenger of Allaah (peace and blessings of Allaah be upon him)
used to say often: “O Controller of the hearts, make my heart steadfast in
adhering to Your religion.” I [Anas] said: “O Messenger of Allaah, we
believe in you and that which you have brought, do you still fear for us?”
He said, “Yes, for people’s hearts are between two of the fingers of the
Most Merciful and He turns them as He wills.” (Narrated by al-Tirmidhi
(2140); classed as saheeh by al-Albaani in Saheeh al-Tirmidhi, 2792). 

We remind you that the Prophet (peace and blessings of
Allaah be upon him) used to like to look on the bright side. It was narrated
from Anas ibn Maalik (may Allaah be pleased with him) that the Prophet
(peace and blessings of Allaah be upon him) said: “(There is) no ‘Adwa (no
contagious disease is conveyed without Allaah’s permission), nor is there
any tiyarah (bad omen from birds). I like fa’l (optimism).” They asked,
“What is fa’l?” He said, “A good word that one of you hears (and takes as a
good omen).” (Narrated by al-Bukhaari, 5776; Muslim, 2224). So be optimistic
that she will remain in Islam. What you have to do now is to strive to guide
her. 

If your wife becomes Muslim then apostatizes,
then in this case it is obligatory to leave her and it is not permissible to
stay with her unless she comes back to Islam. We ask Allaah to end our lives
with righteous deeds.