I know some gods are rather young, mortals from hundreds of years ago that ascended, but I assume Lathander is much older, does it ever state his origins?
Very Old, at least going back to the fall of the Netheril empire
Faiths and Avatars states (page 4):
Long before the Time of Troubles, Ao evidently created some of the powers of the Realms, as well as the crystal sphere of Realmspace. However, he left the Realms powers to change and evolve for millennia untold before deciding that he needed to readjust the balance of the Realms and set in motion the events of the Time of Troubles.
So the oldest powers are literally as old as the world, and existed for many thousands of years.
Lathander is probably at least as old as the fall of Netheril, as some believe that he used to be Amaunator, who died during this event that destroyed the Weave and altered the pantheon (-339 DR):
Over the centuries, many theories have been put forward by later scholars as to what ultimate fate Amaunator met. Some believe he was either absorbed into or became Lathander
However, he could be a lot older (ibid p. 44):
Before the Dawn Cataclysm, there was but one goddess of luck, Tyche. Ever flirting with fortune and disaster, Lady Luck bestowed and withdrew her favor at a whim. After eons of toying with the affections of various powers, Tyche found herself embroiled, along with the rest of her circle of deities, in a war between the gods resulting from the actions of a current paramour, Lathander.
which suggests that he could already have existed for "eons" before that event, although it is not conclusive that he was among her consorts at a much earlier time. The Dawn Cataclysm dates some time after the fall of Netheril (ibid, p. 23), where Lathander is described as still being youthful:
The Dawn Cataclysm, known to have occurred after the fall of Netheril, is believed to have heralded the fall of Myth Drannor. This Godswar is believed to have been sparked by a naive attempt by the then-youthful deity Lathander to remake the Realms from scratch
Th Age of Men is said to begin about -3000 DR with the rise of the Netheril Empire, and Lathander is depicted as a young, atttractive man, which, if he was created by the belief of these humans, would put his creation around that time. If he was created by Eo, he of course could be a lot older still, maybe venerated in different forms and under different names. The Forgotten Realms Campaign Setting for 3e states:
While usually shown as a young, attractive man, Lathander is an old power with a long history of driving creation, progress, and innovation.
So, while I was not able to find a confirmed date for his origin (unless he is the transformed Aumunator), he is at least over one-and-a-half thousand years old, and probably much, much older.