Fasting one day in Rajab is like fasting a hundred years?

No, it is rejected.

This Hadith has been pointed out by many scholars as being inauthentic.

في رجب يوم وليلة، من صام ذلك اليوم، وقام تلك الليلة كان كمن صام من الدهر مائة سنة، وقام مائة سنة، وهو لثلاث بقين من رجب، وفيه بعث، الله محمدًا. هذا حديث منكر إلى الغاية. (تبيين العجب بما ورد في شهر رجب)

There is a night and a day in Rajab, whoever fasts that day and stays up worshiping during the night, will be like a person who fasts a hundred years, and stays up worshipping a hundred years. That night is when three days remain in the month of Rajab, and in it Allah (Most High) sent (the Prophet ﷺ) Muhammad. This is an extremely Munkar (rejected) Hadith. (Imam Ibn Hajr al-Asqalaani, Tabyeen al-Ajab).

Imam al-Sayuti mentioned this narration and then specified one of the narrators as being rejected, many scholars quoted Imam al-Sayuti when commenting on this Hadith.

هيّاج تركوا حديثه (الزيادات على الموضوعات)

They (the Hadith scholars) rejected the narrations of Hayyaaj. (Imam al-Sayuti, al-Ziyaadaat ala -al-Mowdo’aat).

قال السيوطى فى ذيل الموضوعات هياج تركوا حديثه (اتحاف السادة المتقين)

(Imam) al-Sayuti said in Dhail al-Mowdo’aat, they (the Hadith scholars) rejected the narrations of Hayyaaj. (Imam Murtada al-Zabidi, Ithaaf al-Saadah al-Mutaqeen).

قَالَ فِي الذَّيْلِ: فِي إسناده: هياج، تركوه.(الفوائد المجموعة)

In al-Dhail he (al-Sayuti) said, the chain has Hayyaaj in it, they (the Hadith scholars) rejected him. (Imam Showkani, al-Fawaaid al-Majmoo’ah).

Despite this Hadith being rejected, it and other similar narrations are narrated and forwarded by many Muslims. This is a grave error and people must take more care when attributing words or actions to the Holy Prophet ﷺ. Anyone who loves the Holy Prophet ﷺ is protective regarding him ﷺ, and part of that is to oppose false narrations being attributed to our beloved Prophet ﷺ.

And Allah (Most High) Knows Best.

– Answered by Shaykh Noorud-deen (18.02.2021)

See also:

Using Weak Hadith