All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
The Hadith" It`s from the signs of the Last Hour that the new moon would appear older than it`s true age " was narrated by Abi Hurairah, Ibn Mas`ood, Anas Bin Malik and Talhah Bin Abi Hadar (May Allah be pleased with them).
However, its Isnad (chain of narrators) isn`t free of error, and this makes it Da`if (Weak Hadith). Actually, it isn`t only Da`if, but also Munkar (Denounced Hadith),so it isn`t permissible to trace it to the Prophet (PBUH). Rather, it is amongst the Morsal Hadiths (Hadiths lacking first transmitter) narrated by Amer Al-Sha`bi, and having different chains of narrators doesn`t make it authentic, because some of them have narrated denounced Hadiths and their situation was unknown. This is in addition to the fact that the text of this Hadith is Ghareeb (Unfamiliar) since the moon is one of Allah`s creatures and He, The Almighty, told us that it moves according to a certain order and that He measured out stages for her, as reflected in the following verses: "It is He Who made the sun to be a shining glory and the moon to be a light (of beauty), and measured out stages for her; that ye might know the number of years and the count (of time). Nowise did God create this but in truth and righteousness. (Thus) doth He explain His Signs in detail, for those who understand."{Younis, 5}, " And the Moon,- We have measured for her mansions (to traverse) till she returns like the old (and withered) lower part of a date-stalk.It is not permitted to the Sun to catch up the Moon, nor can the Night outstrip the Day: Each (just) swims along in (its own) orbit (according to Law)."{Ya-Seen, 39-40} and "The sun and the moon follow courses (exactly) computed ;"{Ar-Rahman, 5}.Therefore, in reality, the moon can`t look older than its true age.
When they see the new moon, on the first day of the lunar month, big, many of the common mistakenly conclude that the day before was the first day of Ramadan, but this is unacceptable, because the beginning of Ramadan is conditioned upon sighting the crescent or the possibility to do so. However, it (Moon) could be born already, but it is impossible to see it due to astronomical conditions, and so it isn`t announced that Ramadan has begun. Appearing big on the first day of the lunar month is expected, because it was born long ago. However, it couldn`t be actually seen or it wasn`t possible to.
In conclusion, it is worth pointing out that the so-called "Hadith" mentioned above is obviously a group of words that has spread amongst people and isn`t traceable to the Prophet (PBUH). And Allah knows best.