It is said that he had a greater lifespan than any other Dwarf awoken in the First Age and was praised by other Dwarves as the eldest of the Dwarven race. Did Tolkien ever specify how many years he lived?
I don't think Tolkien specified his age but we can nevertheless calculate an upper limit.
From the Apendix A of LOTR, section III Durin's folk:
There he lived so long that he was known far and wide as Durin the Deathless. Yet in the end he died before the Elder Days had passed, and his tomb was in Khazad-dûm; but his line never failed, and five times an heir was born in his House so like to his Forefather that he received the name of Durin.
So we know that Durin awoke Mount Gundabad in the Years of the Trees (I think this is better than calculating the time he spent in sleep after his Creation) and we also know that Durin died in the First Age.
From the Annals of Aman and The Grey Annals (History of Middle Eearth X and XI) we know that one year of the trees is roughly 10 sun years and that the Elves were discovered in 1085 by Oröme in Cuiviénen:
But as for the Years of the Trees and those that came after, one such year was longer than nine such years as now are. For there were in each such Year twelve thousand hours. Yet the hours of the Trees were each seven times as long as is one hour of a full-day upon Middle-earth from sun-rise to sun-rise, when light and dark are equally divided. Therefore each Day of the Valar endured for four and eighty of our hours, and each Year for four and eighty thousand: which is as much as three thousand and five hundred of our days, and is somewhat more than are nine and one half of our years (nine and one half and eight hundredths and yet a little).
Valian Year(s), S.Y. = Sun Year(s)): V.Y. 1000 = S.Y. 10000 First flowering of the Trees V.Y. 3000 = S.Y. 30000 Rising of the Moon
Assuming that Durin was already awoken when Örome found the Eldar we can calculate that he could live at most 19150 years in the Ages of the trees. To that amount of years we should add all the years of the First Age of the Sun, again assuming the upper limit and considering that Durin died in the last day of the First Age. The First Age lasted 587 years from the rise of the sun to the ruin of Beleriand after the War of Wrath.
Concluding: Years of the Trees (19150) + First Age (587)= 19737 That's the roughly calculated upper limit for Durin I age.