It is certainly the duty of every Muslim to try to propagate Islam and convey its message to the people. The whole Muslim community is responsible for making Islam known to the rest of mankind and calling on them to embrace it as a faith and a way of life. In addition, every Muslim should do their best to propagate Islam among non-Muslims and to encourage Muslims to observe their religious duties. This was the task of prophets and it was brought to its fullness by Prophet Muhammad, the last Messenger (peace and blessings be upon him). He spared no effort in explaining God’s message to mankind and he worked hard for his followers to establish a base for Islam from where it spread to vast areas of world.
One important point to remember about Islam is its total approach to human life. It does not emphasize any single aspect at the expense of another. It does not tolerate the neglect of a duty only because a person is happier fulfilling another. The needs of this world must not be neglected because one wishes to devote oneself to the fulfillment of duties relevant to the life-to-come. Islam provides for us a middle ground which takes care of all man’s needs as individuals, members of a family, and active participants in the community. In the Quran we read this injunction which applies to every single one of us: "Seek, by means of what Allah has granted you, [the good of] the life to come, without forgetting your own rightful share in this world; and do good to others as Allah has done good to you" [28: 77].
All of us know the well-known story of the three men who went to the wives of the Prophet (peace and blessings be upon him) to inquire how he worshipped [God]. They said that this is only to be expected because the Prophet (peace and blessings be upon him) had been forgiven his past and future sins and that they cannot aspire to that. Therefore, they have to make a greater effort to win God’s pleasure. One of them pledged to spend the night in worship, the second pledged to fast every day of his life and the third pledged permanent celibacy. When the Prophet (peace and blessings be upon him) learned of this, he reproached them. He told his Companions: “I am the most God-fearing among you. I spend part of the night in worship but also allow myself enough time to sleep, I perform voluntary fasts but there are many days when I do not fast, and I also marry women… He who abstains from following my example is not from me [not one of my followers].”
When we look at the Prophet’s life, we realize that he never neglected a duty. When it was time to go to war for the cause of Islam, he did not hesitate to march at the head of his army. He never neglected something which he felt would give pleasure to one [or more] of his wives provided he was able to give it for her.
Your friend has certainly many duties toward his family. He is responsible for their maintenance and he has to work hard to provide his wife and children with the best standard of living he can afford. If he does not, then he is guilty of neglecting his duty. No Muslim is deemed to have discharged his responsibility toward his wife and children if he chooses to stay without a job. Moreover, his duty toward his family far exceeds the provision of food and clothing and shelter. He must look after the upbringing of his children and make sure that they grow up to be good Muslims. If he neglects this duty to serve what seems to him to be a higher or nobler duty, he will be accountable to God the Almighty for this neglect. Indeed, taking good care of one’s children is, in Islam, a primary duty and takes precedence over many other duties.
Your relative's attitude is highly reprehensible. He neglects the education and upbringing of his own children to do something which God does not require him to do in this particular fashion. His duty to propagate Islam would be better served by raising his children as good Muslims. To say that he entrusts his family to God’s care is good enough but only if he has compelling reasons for his absence. But the reason he gives is not a compelling one because he can propagate Islam from his home. Why should he go abroad for a year to do this?
From another point of view, a man's duty toward his wife is such that he is not allowed to be absent from his home for more than four months unless she willingly agrees to his absence. Umar Ibn Al-Khattab issued a general decree preventing soldier from staying away from their wives for more than four months. He did not want Muslim wives to suffer the long absences of their husbands. If such long absences are unacceptable when a person is going on jihad, how can they be acceptable when he is engaged in an activity which can be, at best, described as desirable or voluntary? It is wrong of your relative to make his wife suffer. He will be better rewarded if he takes good care of his family and tries to propagate Islam and its message from his home.