Several of the characters in our group are of long-lived races and have ages in the triple digits as a result. Presumably the game-mechanic effects of their ages are already factored into the rules for playing the race; any advice on how to role-play them, though? How does one go about role-playing a level-3 character who is more than a hundred years senior to the rest of the team but at the same in-game skill level?
Long lived characters would also have had lots of time to get things done, so might take a lot longer to consider their response to challenges than the short lived races. Such a character would either feel angry/irritable when being "rushed" or would just not commit to making a decision until they had a "reasonable" amount of time to consider the issue at hand. In combat they might be swift and deadly, but when planning how to storm the citadel may want to take their time to look at it from every direction.
In these situations, I look to fiction - especially film. Stories of characters that have lived unusually long - Anything modern with vampires, or the older Highlander (Sean Connery!) stories, for example. There is both much wisdom and great tragedy in out-living other races - more specifically and the people you love and hate.
Look for the quirky behaviors and attitudes exhibited by these long-lived characters: Are they unusually quiet and contemplative. Do they get easily disgusted with the same old "childish" behavior they've seen over in over?
I'm reminded of a line by Merlin (who lives a long time, and backward no-less) in 1970s film Excalibur during the Siege of Camelyarde: When the young page Arthur "the boy king" hands Excalibur to his enemy in order that he may may in-turn make Arthur a knight, when Merlin remarks: "I never saw this!" with great delight. Man, I'd so use that ham-fisted Merlin characterization the first chance I got... :-)
With many long-lived races, those extra years are spent in isolation, either in some idyllic fairyland where little changes (as with elves) or in torpid slumber as the years pass (as with vampires). You can portray this by being a little spacey or out-of-touch with the rest of the world.
Very-long-lived characters make decisions quickly. They have already been in most conceivable circumstances, and have made their errors of youth long ago... but they may thus, in their inertia, devalue the opinions of others ("Been there done that, I'm right you're wrong").
Look carefully at Heinlein's classic Time Enough for Love, especially the two interludes featuring his quotations. (Caution: Woodie/Lazarus is quirky, and it takes some skill for the reader to separate the personality from the age-effects.) Like Woodie, a character should keep a watchful eye on the broader view, anticipating situations and having plans worked out for many contingencies.
Rule One is to stay alive. All other things -- fame & fortune, loot & equipment, even friendships (especially with the short-lived) -- are secondary.
One way to reflect a long life apparently spent doing nothing terribly self-improving is idiosyncratic life philosophy; many years for contemplation may have led a character to place an unusually high value on certain behaviors or to object unusually strenuously to others.
There's a general presumption that very long-lived types must be stick-in-the-ass conservatives. Journey to the West: The Monkey King provides a nice antidote to this, where the monkey king, Nanduka, gains immortality through trickery and gluttony, and turns heaven upside-down with his foolish pride and mischief. He does, sadly, become a reformed character. If you are after stick-in-the ass immortals, it's hard to beat Gloranthan Brithini.
There's no obligation to conform to stereotypes, only to play characters that work for you and the gamers you play with. Talk over your ideas with them until you've got an idea you are happy about.