Assalamualaikum ustaz. When is the last third of the night? From what I understand, 7-12 p.m. is considered night time and 1-6 a.m. is considered as the morning. And when is it can we perform tahajjud prayer?
Alhamdulillah, praise and gratitude to Allah SWT for His countless blessings for us all. Praise and salutations to our beloved Prophet Muhammad PBUH, his family, companions and all those who follow his footsteps until the Last Day.
The beginning of the night is when the time of Maghrib prayer starts and ends when the Isyak time ends. If a person is in Kuala Lumpur and the time for prayer considered is according to the 24-hour system, the time for Maghrib starts at 19:21 and the time for Isyak ends at 05:52. Then, the total duration of the night is 10 hours and 50 minutes. If one-third of the night is 3 hours and 50 minutes, then the starting of the last third of the night is approximately at 02:21 until the time for Subuh.
Regarding Tahajjud prayer, this prayer is performed after one wake up from sleep and before the time for Subuh.  From Abu Hurairah RA, Rasullullah PBUH said:
يَنْزِلُ رَبُّنَا تَبَارَكَ وَتَعَالَى كُلَّ لَيْلَةٍ إِلَى السَّمَاءِ الدُّنْيَا حِينَ يَبْقَى ثُلُثُ اللَّيْلِ الآخِرُ يَقُولُ: مَنْ يَدْعُونِي، فَأَسْتَجِيبَ لَهُ مَنْ يَسْأَلُنِي فَأُعْطِيَهُ، مَنْ يَسْتَغْفِرُنِي فَأَغْفِرَ لَهُ
Our Lord who is blessed and exalted descends every night to the lowest heaven when two-thirds of the night has passed and says, “Who supplicates me so that I may answer him? Who asks of me so that I may give to him? Who asks my forgiveness so that I may forgive him?” 
Syeikh Mustafa al-Bugha said: (Including) what is meant by “descends” is the descent of His Will, Blessings and Tenderness.
Tahajjud prayer has great advantages as stated in a hadith narrated from Abu Hurairah RA, the Prophet PBUH said:
وَأَفْضَلُ الصَّلَاةِ بَعْدَ الْفَرِيضَةِ؛ صَلَاةُ اللَّيْلِ
The most excellent prayer after that which is obligatory is the (voluntary) late-night prayer.” 
Ibn Rajab said: “The time for tahajjud during the night is the best time to perform sunnah prayer. At the time, a slave is close to His Lord. The time is when the doors of the heavens are opened and supplications are granted. This is also the time to ask the various matters we want to Allah SWT.” 
Lastly, we are inclined towards the division of the night as just taqribi (estimation or approximation) and not tahdidi (accurate). It is up to each individual to perform qiamullail according to one’s own comfort and capability.
May Allah SWT grant us love to perform night prayers and Tahajjud and make us from amongst those who are praised in the hereafter. Allah SWT states:
وَمِنَ اللَّيْلِ فَتَهَجَّدْ بِهِ نَافِلَةً لَّكَ عَسَىٰ أَن يَبْعَثَكَ رَبُّكَ مَقَامًا مَّحْمُودًا
“And from [part of] the night, pray with it as additional [worship] for you; it is expected that your Lord will resurrect you to a praised station.”
Surah al-Isra’ (79)