All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is obligatory for a Muslim to perform the prescribed prayers in the time that is determined for them by the sharia. Allah Almighty says (what means): {Indeed, prayer has been decreed upon the believers a decree of specified times.} [Quran 4:103]
Performing the prayer at their prescribed times is also enjoined by the Sunnah. It was narrated on the authority of ‘Ubadah ibn As-Samit that he said:
I bear witness that I heard the Messenger of Allah, sallallaahu ‘alayhi wa sallam, say, “Allah, the Exalted, has enjoined five daily prayers: whoever performs ablution for them properly, prays them on time, bows properly in them, and focuses with proper submissiveness (khushoo‘), has a promise from Allah that He will forgive him. And whoever does not do that has no such promise from Allah; if He wills, He will forgive him and if He wills, He will punish him.” [Malik]
Staying up late after the ‘Isha’ prayer is disliked according to the scholars. It was narrated on the authority of Abu Barzah that he said, “The Messenger of Allah, sallallaahu ‘alayhi wa sallam, disliked sleeping before the ‘Isha’ and speaking thereafter.” [Al-Bukhari]
If one knows that staying up late at night will cause him to miss some prayers and that the alarm clock or whatsoever means will be of no benefit, then it is prohibited for him to stay up that late. This is so because matters are judged based upon their purposes, and what leads to a prohibition is in itself prohibited. The hadith of the lifting of the pen upon which that person bases his claim is an authentic hadith that was narrated by Abu Dawood, An-Nasa‘i, At-Tirmithi and Ibn Majah. However, it cannot be used as evidence in such a case because lifting the pen for the sleeping person until he wakes up does not entail the permissibility of staying up late when it leads to such consequences.
Allah knows best.