What is indicated by the texts of the Qur’an and Sunnah, and the consensus of the early generations of this ummah is that no one who has an atom’s weight of faith in his heart will abide in Hell for eternity. If a Muslim who was a sinner died without repenting from his sin, then his case will be decided by Allah: if He wills He will pardon him, and if He wills He will punish him, but he will not abide in Hell forever in any case. 

Al-Bukhaari (44) and Muslim (193) narrated from Anas that the Prophet (blessings and peace of Allah be upon him) said: “Whoever says Laa ilaaha ill-Allah and has in his heart goodness the weight of a grain of barley will be brought out of Hell, then whoever says Laa ilaaha ill-Allah and has in his heart goodness the weight of a grain of wheat will be brought out of Hell, then whoever says Laa ilaaha ill-Allah and has in his heart goodness the weight of an atom (or a small ant) will be brought out of Hell.” 

This idea is repeated in other hadeeths, the wording of which is similar. 

With regard to your saying that no one who has an atom’s weight of faith in his heart will enter Hell, this is also proven, but some of the scholars interpret it as meaning that such a one will not enter it as the disbelievers do, i.e., he will not abide therein for eternity, although he will may enter it. Thus this hadeeth may be reconciled with the many texts that say that some of the sinners among the believers will enter Hell even though they have faith in their hearts, then they will be brought out by means of intercession and so on. 

At-Tirmidhi (1999), Abu Dawood (4091) and Ibn Maajah (59) narrated from ‘Abdullah ibn Mas‘ood that the Prophet (blessings and peace of Allah be upon him) said: “No one will enter Paradise in whose heart is an atom’s weight of arrogance and no one will enter Hell in whose heart is an atom’s weight of faith.” A man said: What if I like my clothes to look nice and my shoes to look nice? He said: “Verily Allah loves beauty; rather arrogance means rejecting the truth and looking down on people.”

Classed as saheeh by al-Albaani in Saheeh at-Tirmidhi. 

Imam at-Tirmidhi (may Allah have mercy on him) said, after quoting this hadeeth: Some of the scholars said, commenting on this hadeeth, that no one who has an atom’s weight of faith in his heart will enter Hell; the only meaning is that they will not abide in the Fire forever. This was narrated from Abu Sa‘eed al-Khudri from the Prophet (blessings and peace of Allah be upon him) who said: “Everyone in whose heart there is an atom’s weight of faith will be brought out of Hell.” More than one of the Taabi‘een interpreted the verse (interpretation of the meaning), “Our Lord! Verily, whomever You admit to the Fire, indeed, You have disgraced him” [Aal ‘Imraan 3:192] as meaning: whomever You cause to abide forever in the Fire, indeed, You have disgraced him. End quote. 

With regard to the second sentence, in which you say, “that the one who has an atom’s weight of arrogance in his heart will not enter Paradise”, this is also proven, as in the hadeeth quoted above. It was quoted by Muslim (91), with this sentence only, from ‘Abdullah ibn Mas‘ood from the Prophet (blessings and peace of Allah be upon him) who said: “No one will enter Paradise in whose heart is an atom’s weight of arrogance.” A man said: What if a man likes his clothes to look nice and his shoes to look nice? He said: “Verily Allah is Beautiful and loves beauty; arrogance means rejecting the truth and looking down on people.” 

What this hadeeth means is that no one will enter Paradise without being exposed to the possibility of requital or punishment; he may be punished or he may be pardoned, unlike the obedient believer who is free of this major sin, who will enter Paradise without being exposed to the possibility of entering Hell.  

An-Nawawi (may Allah have mercy on him) said in Sharh Muslim: Rather the apparent meaning is that which was favoured by al-Qaadi ‘Iyaad and other scholars, which is that he will not enter Paradise without bringing him to account and punishing him if He wills, or He may let him off. Rather it is inevitable that all of those who affirm the Oneness of Allah will enter Paradise, either initially or later on, after some of those who committed major sins and died whilst persisting in them have been punished. And it was said that he will not enter Paradise with the pious in the first group to enter.

End quote. 

To sum up: 

What you asked about is the text of one or two saheeh hadeeths, the meaning of which is as we have said. It is not correct for anyone to imagine that it is impossible for the one who has an atom’s weight of faith in his heart to enter Hell no matter what major sins he has committed, and he should not imagine that it is impossible for the Muslim in whose heart there is an atom’s weight of arrogance to enter Paradise, no matter how great his faith. 

Two groups of people did not understand these texts corectly, the Khaarijis and the Murji’ah. The Khaarijis focused only on the texts that contain the warning, such as the second hadeeth, and the Mujri’ah focused only on the texts that contain the promise, such as the first hadeeth. But Allah guided Ahl as-Sunnah wa’l-Jamaa‘ah to the correct view, which reconciles between the texts. 

And Allah knows best.