Get used to being in old city quarter (5)
I believe the answer is:
inure
'get used to' is the definition.
(I've seen this in another clue)
'in old city quarter' is the wordplay.
'old city' becomes 'ur' (ancient city state).
'quarter' becomes 'e' ().
'in'+'ur'+'e'='INURE'
'being' is the link.
(Other definitions for inure that I've seen before include "Harden or accustom as to something unpleasant" , "Come to accept" , "Cause to become hardened to" , "Get used to conditions" , "get tougher" .)