Admittedly that quotation does not in itself indicate what Smaug means by 'of old'. In its proper context it follows the dragon's comments about those likely to want revenge on him: "Revenge! The King under the Mountain is dead and where are his kin that dare seek revenge? Girion Lord of Dale is dead, and I have eaten his people like a wolf among sheep, and where are his sons' sons that dare approach me?". Although this seems to me to imply that 'the warriors of old' are those he killed in taking Erebor, the real key to his age comes a little earlier in the chapter:
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And Smaug laughed aloud. He had a wicked and a wily heart, and he knew his guesses were not far out, though he suspected that the Lake-men were at the back of the plans, and that most of the plunder was meant to stop there in the town by the shore that in his young days had been called Esgaroth.
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Earlier still, Tolkien wrote of the Lake-men:
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They still throve on the trade that came up the great river from the South and was carted past the falls to their town; but in the great days of old, when Dale in the North was rich and prosperous, they had been wealthy and powerful
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'Days of old' in this context means, then, the time before the coming of Smaug. In any case, I would think it unusual that a dragon so ancient as to have escaped from Angband's destruction had no more in his hoard than what he had been able to steal from Thror and the men of Dale. For me, all of the evidence points to Smaug being somewhere in the region of two-hundred years old, probably slightly older; and only a descendant of the dragons of the First Age as Saucepan suggested.