Yes, Fasting is Shaban is Sunnah of prophet Muhammad PBUH Reference from Ibn e Majah.
The month of Shaban is the eighth month of the lunar year in the Islamic calendar.
And the month immediately preceding the month of Ramadan.
For Muslims all around the world love to fasting in Shaban, the month of Shaban is a special time in which,
they try to do more good deeds than usual and fast more optional fasts than they do in other months.
The Prophet Muhammad (PBUH) used to fast a lot during this month, he used to keep fasting in Shaban.
It is the habit of the Muslims to celebrate the 15th night of the month of Shaban by praying, reciting Quran, Watch Islamic TV Online
praising Allah, and making a great deal of supplication to Allah during that night. Ibn Majah narrated that the Prophet said:
“قال رسول الله:
“اذا كانت ليلة النصف من شعبان فقوموا ليلها وصوموا نهارها”
رواه ابن ماجة.
which means:
“Spend the night preceding the 15th day of Shaban in acts of obedience, and fast the next day.
“ The acts of obedience meant in this hadith are reciting the Quran, making supplication (dua) to Allah, and praying optional prayers.

Even though there is a difference in opinion among the top scholars of hadith about the degree of confirmation of this hadith,
acting in accordance with its orders (on that particular night, i.e., the 15th night of Shaban) is a good practice.
This is so because the matters it orders with (i.e., reciting Qur’an, making supplication, praising Allah) are, in general, recommended matters in the Religion.
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