Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
There is no harm for the said man to agree with his parents and accept to marry the woman his parents proposed him, even if she is older than him or a widow, because that does not decrease anything from her dignity and honour provided she is a pious woman with good manners, and is chaste.
A wife is chosen according to four characteristics mentioned in an authentic Hadith reported by Bukhari and Muslim. The Prophet (Sallallahu Alaihi wa Sallam) said: “A woman is married for her wealth, her honourable lineage (family status), her beauty, and her religion. Opt for the religious woman, otherwise you will be a loser.”
As regards the relationship between him and a non-Muslim woman or any other non-Mahram woman, it is a great and major sin. He should seek repentance from this, regret that act, and be determined not to return to it, because Allah forbade a person to have any relationship of this kind outside of marriage. Therefore, it is incumbent on him to, first and foremost, cut relations with this woman.
However, if she becomes a Muslim and he likes to marry her, then it is permissible for him to do so under certain conditions.
It is also permissible for him to marry her if she is a chaste Christian, but a Muslim woman is better for him.
Allah knows best.