Office admirer going round old city (6)
I believe the answer is:
bureau
'office' is the definition.
('bureau' can be a synonym of 'office')
'admirer going round old city' is the wordplay.
'admirer' becomes 'beau' (I've seen this before).
'going round' indicates putting letters inside.
'old city' becomes 'ur' (ancient city state).
'beau' enclosing 'ur' is 'BUREAU'.
(Other definitions for bureau that I've seen before include "Writing desk with drawers" , "Agency - writing desk" , "Government department, agency" , "Ancient chipped flint" , "French office - desk" .)