Arahants experience ageing and death but do not suffer from it, after attaining Nibbana.
From DN 16:
"Now I am frail, Ananda, old, aged, far gone in years. This is my
eightieth year, and my life is spent. Even as an old cart, Ananda, is
held together with much difficulty, so the body of the Tathagata is
kept going only with supports. It is, Ananda, only when the Tathagata,
disregarding external objects, with the cessation of certain feelings,
attains to and abides in the signless concentration of mind, that his
body is more comfortable.
Experience of sensations continue to occur for arahants till their parinibbana, as stated in Iti 44:
And what is the Unbinding property with fuel remaining? There is the
case where a monk is an arahant whose fermentations have ended, who
has reached fulfillment, finished the task, laid down the burden,
attained the true goal, ended the fetter of becoming, and is released
through right gnosis. His five sense faculties still remain and, owing
to their being intact, he is cognizant of the agreeable & the
disagreeable, and is sensitive to pleasure & pain. His ending of
passion, aversion, & delusion is termed the Unbinding property with
fuel remaining.
And what is the Unbinding property with no fuel remaining? There is
the case where a monk is an arahant whose fermentations have ended,
who has reached fulfillment, finished the task, laid down the burden,
attained the true goal, ended the fetter of becoming, and is released
through right gnosis. For him, all that is sensed, being unrelished,
will grow cold right here. This is termed the Unbinding property with
no fuel remaining."
Commentary (Thanissaro):
With fuel remaining (sa-upadisesa) and
with no fuel remaining (anupadisesa): The analogy here is to a fire.
In the first case, the flames are out, but the embers are still
glowing. In the second, the fire is so thoroughly out that the embers
have grown cold. The "fuel" here is the five aggregates. While the
arahant is still alive, he/she still experiences the five aggregates,
but they do not burn with the fires of passion, aversion, or delusion.
When the arahant passes away, there is no longer any experience of
aggregates here or anywhere else.
But what does it mean when we say that Arahants become deathless? This is because of Nibbana, suffering ends, birth ends and so death also ends.
From Dhammapada 400:
Akkodhanam vatavantam
silavantam anussadam
dantam antimasariram1
tamaham brumi brahmanam
Verse 400: Him I call a brahmana, who is free from anger, who
practises austerity, who is virtuous and free from craving, who is
controlled in his senses and for whom this body (i.e., existence) is
the very last.
Footnote 1. antimasariram: lit., one who has the last body. This is
his last body because he will not be reborn; he is an arahat.
But how does the Arahant become free from suffering and death? As we know from the third noble truth, suffering is ceased when craving is ceased. The requisite condition for ageing and death is birth.
From SN 44.9:
And at the moment when a being sets this body aside and is not yet
reborn in another body, what do you designate as its sustenance then?"
"Vaccha, when a being sets this body aside and is not yet reborn in
another body, I designate it as craving-sustained, for craving is its
sustenance at that time."