The hadeeth (narration) you mention is a saheeh (authentic) hadeeth. It was narrated by Muslim, but there is a difference of scholarly opinion as to the meaning of the word “fa aslama” in this hadeeth. This dispute and the most correct view was mentioned by al-Nawawi in his commentary on this hadeeth. He said: 

With regard to the words of the Prophet (peace and blessings of Allah be upon him): “There is none of you who does not have a companion (qareen) appointed for him from among the jinn.” They said, “Even you?” He said, “Even me, but Allah helped me against him fa aslama [or fa aslamu], so he only tells me to do that which is good.” 

There are two well-known views that have been reported. Those who read the phrase as fa aslamu said that it means, “So I am safe [aslamu] from his evil and temptation.” Those who read it as fa aslama said that it means, “The qareen became Muslim [aslama] and became a believer, so he only tells me to do that which is good.” 

They differed as to which view is correct. Al-Khattabi said: The correct version is fa aslamu [so I am safe]. Al-Qadi ‘Iyad thought that fa aslama [so he became Muslim] was correct, and this is the preferred version, because he then said, “so he only tells me to do that which is good.” And they differed concerning the report that says fa aslama. It was said that it means he submitted in the sense of surrendering, and it appears in this form (fa astaslama – so he surrendered) in reports narrated elsewhere than in Saheeh Muslim. And it was said that it means that he became a Muslim and a believer. This is the apparent meaning. 

Abu Na’eem al-Asbahani said in Dalail al-Nubuwwah: It was said aslama meaning he believed. The Prophet (peace and blessings of Allah be upon him) was the only one whose qareen became a Muslim and a believer. 

Based on this, having one’s qareen become a Muslim was something that was unique to the Prophet (peace and blessings of Allah be upon him). 

So it is not prescribed for the Muslim to pray that his qareen become Muslim, because this is overstepping the limits in du’a (supplication) by asking Allah for something that belongs only to the Prophet (peace and blessings of Allah be upon him), and because there is no report that the sahabah (companions), who were the keenest of all people to do good and the closest to it, asked Allah to make their qareens become Muslim, and they did not ask the Messenger (peace and blessings of Allah be upon him) to pray for that for them when they heard this hadeeth from him. There is no report that Abu Bakr, ‘Umar, ‘Uthman and ‘Ali (may Allah be pleased with them), who were very keen to do good, or their sons, did that. We have to follow the guidance of these great sahabah because they understood this religion properly and learned it directly from its source, the Messenger of Allah (peace and blessings of Allah be upon him), and it is not permissible for us to follow any path but theirs. Allah says (interpretation of the meaning): 

“And whoever contradicts and opposes the Messenger after the right path has been shown clearly to him, and follows other than the believers’ way, We shall keep him in the path he has chosen, and burn him in Hell — what an evil destination!” [4:115] 

This hadeeth was narrated in the context of warning the sahabah against the fitnah (trials) of the qareen. Al-Nawawi said: This hadeeth is a warning against the fitnah of the qareen and his whispers and temptation. [The Prophet (peace and blessings of Allah be upon him)] warned us against him so that we can be as cautious as possible. This is what we must do, according to sharee’ah (Islamic law). 

It is sufficient for us to recite the du’as that Allah has taught us, for example: 

“And say: ‘My Lord! I seek refuge with You from the whisperings (suggestions) of the shayateen (devils).

And I seek refuge with You, My Lord! Lest they should come near me.’” [23:97-98 – interpretation of the meaning] 

And you should recite Surat al-Ikhlas and the Mu’awwadhatayn (the last two surahs of the Quran) morning and evening, as well as the dhikrs (words of remembrance) narrated in saheeh ahadeeth (authentic narrations) from the Prophet (peace and blessings of Allah be upon him). You should also recite ayat al-Kursiy before going to sleep, and Bismillahi al-Rahmani al-Raheem before doing anything, and you should seek refuge with Allah every time you notice the whispers of the shaytan. Allah says (interpretation of the meaning): 

“And if an evil whisper from shaytan  tries to turn you away (O Muhammad) (from doing good), then seek refuge in Allah. Verily, He is the All‑Hearer, the All‑Knower.” [41:36] 

I ask Allah to make you more keen to do good and to protect us and you from the shaytan and his whisperings. Ameen.