Some Tartarians exit the gate with old age and they most certainly get the old age marker and are affected by the stat reductions of old age despite being undead.
From a practical perspective, most undead have so long a "life" ahead of them if you check their age on the fatigue summary that it is unlikely in the extreme that you'll ever see them as old if they aren't already old when you get them (you'd have to play many, many, thousands of turns), but exceptions do exist as the Tartarian example shows (due to them only getting a max age of 500 and starting with 400-600 age, whereas e.g. Dusk Elders typically have a max age of 2500-4500 depending on how much you boost their death paths and start at something like 400-600 too. Yeah, those ancient dead gods are puny youngsters compared to the dusk elders
)