I wanted to know that prophet muhammad used to pray the namaz in the night, i think it is called tawju namaz, but how did he still pray fajr. Does that mean that he used to sleep for about 3-4 hours? 

In the name of Allah, Most Compassionate, Most Merciful, 

AbuUmamah narrates that Allah’s Messenger (sallal lahu alayhi wa sallam) said: Make it a point to observe Tahajjud prayer, for it was the practice of the pious gone before you (and it is the means whereby you can find) nearness to your Lord and (seek) expiation for the sins and (find) prevention from evil. (Tirmidhi: 362)

From the above and similar ahadiths the importance and reward of tahajjud is clear. It was the practice of the Prophet (sallal lahu alayhi wa sallam) to offer tahajjud prayer every night.

Masruq narrates that I asked ‘Aisha “What deed was the most beloved to the Prophet?” She said, “The regular constant one.” I said, “At what time did he use to get up at night (for the Tahajjud night prayer)?” She said, “He used to get up on hearing (the crowing of) the cock (the last third of the night).” (Sahih Bukhari: 8.468)

The practice of the Prophet (sallal lahu alayhi wa sallam) was to offer tahajjud in the last third of the night just before suhoor. If a person goes to bed straight after Esha prayers then he would have enough time to sleep and still get up for tahajjud and Fajr prayers. Also a person can take a siesta during the day as was the practice of the Prophet (sallal lahu alayhi wa sallam).

Umm Sulaym narrates that Allah’s Apostle (peace be upon him) visited her house and (took rest) and she spread a piece of cloth for him and he had a siesta on it… (Sahih Muslim: 1084)

Only Allah knows best

Sayeedur Rahman

Darul Ifta, Birmingham