Self-centred person, say, is absorbed in old Times (7)
I believe the answer is:
egotist
'self-centred person' is the definition.
(I've seen this before)
'say is absorbed in old times' is the wordplay.
'say' becomes 'eg' (short for 'exempli gratia', 'for example' in Latin).
'absorbed in' means one lot of letters goes inside another.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'times' becomes 'tt' (two abbreviations for time).
'o'+'tt'='ott'
'is' placed inside 'ott' is 'otist'.
'eg'+'otist'='EGOTIST'
(Other definitions for egotist that I've seen before include "Self-obsessed individual" , "Conceited type" , "One who is concerned only with self" , "Boastful person" , "show-off" .)