Standing for Prayer at Night is not specifically for Ramadhan
Question :
Is the standing for prayer at night only done in the blessed month of Ramadhan or during all the nights of the year? From what time does it begin and at what time does it end? Is the standing for prayer only or is it prayer and recitation of the Noble Qur'an?
Answer:
Standing at night for prayer and At-Tahajjud is Sunnah and a virtuous act that the Prophet and his Companions strictly guarded. This is as Allah said:
"Verily, your Lord knows that you stand less than two thirds of the night (in prayer), and half of it, and a third of it, and also of group of those who are with you."
This is not specifically for the month of Ramadhan only. Its time is between the times of Al-'Isha' and Al-Fajr. However, the prayer during the last part of the night is better, and if he prays in the middle of the night he will be rewarded. It is better for it to be performed after some sleeping or in the last half of the night. And Allah knows best.
Source:
Ash-Shaykh Ibn Jibreen
Fatawa Islamiyah, Vol. 3 Pages 322-323