Let's see..
In Legends, the "Gods" of Zakuul were, what, thousands of years old at least?
The Abominor and Selentium, not sure if they count as droids, about as old as the Yuuzhan Vong.
There should be lots that are a few hundred years old, but none come to mind.
In Canon, the "Aphra" comics had a bunch of old stuff, because she's an archeologist. I'm just gonna name the easy one: 0-0-0: he's well over a hundred, maybe two hundred, I think.
The "Sentinels" in Fortress Corvax from Vader Immortal have to be at least a few hundred years old, for whatever they're worth.
I suspected C-3PO would be kind of old, considering he was still in 45 ABY (in the Crucible novel), but his earliest appearance, when Anakin found him, was in 32 BBY, with the implication being that he was older than even that.
Wookieepedia linked an article from StarWars.com called "How old is C-3PO, really?" which is now a defunct link, and it claims that he was made 80 years before Anakin found him in 112 BBY on Affa.
But I found the old Los Angeles Times newspaper strips from 1980, and although they are S-Canon, there's a line in which C-3PO says "You're the very best friend I've had, in all the centuries since they assembled me!" to R2.
So, he could qualify, maybe.