(Note that this seems to be a general Forgotten Realms question, rather than being specifically about Living Forgotten Realms, which is WotC's Organized Play campaign for 4E, but that's OK, because I know where to find the answer...)
The
Time of Troubles occurred in DR 1358. In 4E, elves are described as living for several hundred years ("Most elves live to be well over 200 years old and remain vigorous almost to the end." -- Player's Handbook, page 41.) On the other hand, the PHB also notes that "Elves mature at about the same rate as humans, but show few effects of age past adulthood."
So, yes, it'd certainly be possible for a "vibrant" elf in 1479 to have been born in 1358 -- he'd be 121 years old, and could probably expect to live for another hundred years or more. But, bear in mind that said elf has been an adult for about 100 years at that point --which, if your character is just starting his adventuring career in 1479, might be a little strange (it'd sort of be the equivalent of a middle-aged human just setting out on a career...maybe not unheard of, but probably unusual).
In short: nothing in the rules says "no", but you should make sure your DM is cool with it.
"Of course [Richard] has a knife. He always has a knife. We all have knives. It's 1183, and we're barbarians!" - Eleanor of Aquitaine, "The Lion in Winter"
