Never realised the League was quite so old!
Regardless, we have only a very foggy idea about when the first champions joined the League, especially regarding people such as Singed, Taric, Sivir, Anivia, Malphite... They might have joined on slowly as the years went by, gradually giving the League more fame and power bit by bit until he City-States could not ignore their influence.
Annie would have to have been one of the later members lore-wise due to her age, but outside of the lore, yes she is one of the oldest champions. Nunu is the same way if you think about it. Still, relating game time to lore time is a very, very foggy area and I wouldn't reccomend it. I don't think that Riot fully expected the game to take off like a roman candle like it did, causing some small discrepancies. At least, that's my theory.