Visit old city that's somewhat grim (4)
I believe the answer is:
dour
'somewhat grim' is the definition.
'dour' can be an answer for 'grim' (synonyms). I am not certain of the 'somewhat' bit.
'visit old city' is the wordplay.
'visit' becomes 'do' ('do' can mean to visit an attraction or landmark).
'old city' becomes 'ur' (ancient city state).
'do'+'ur'='DOUR'
'that's' is the link.
Can you help me to learn more?
(Other definitions for dour that I've seen before include "Gloomy or sullen" , "Grim and gloomy-looking" , "Severe, gloomy-looking" , "Uncompromising" , "Stern and severe looking" .)