Accusative of respect: 'He's old/an old man with respect to his hair(s)' – i.e., his hair is that of an old man.
Draconis has alluded to this in the other answer, but it's worth making explicit that τὰς φρένας in the last line is the same sort of accusative (whatever name one calls it by): 'He's young in respect to his mind/heart' – i.e., he's young at heart. (Note that the verb νεάζω doesn't mean 'to renew' but 'to be young/new': it's intransitive.)
For the sake of completeness, here are the relevant bits from Smyth, Greek grammar:
1600. To verbs denoting a state, and to adjectives, an accusative may be added to denote a thing in respect to which the verb or adjective is limited.
a. The accusative usually expresses a local relation or the instrument. The word restricted by the accusative usually denots like or similar to, good or better, bad or worse, a physical or a mental quality, or an emotion.
1601. The accusative of respect is employed
a. Of parts of the body: ὁ ἄνθρωπος τὸν δάκτυλον ἀλγεῖ the man has a pain in his finger P. R. 462 d, τυφλὸς τά τ' ὦτα τόν τε νοῦν τά τ' ὄμματ' εἶ blind art thou in ears, and mind, and eyes S. O. T. 371, πόδας ὠκὺς Ἀχιλλεύς Hom.
1602. Very rarely after substantives: χεῖρας αἰχμητής a warrior valiant with (thy) arm π 242, νεανίαι τὰς ὄψεις youths by their appearance L. 10.29.
(Although γέρων is usually a noun, it can also be used as an adjective. In any case, τρίχας γέρων is covered by either §1600 or §1602.)