Catches up with an old Greek (7)
I believe the answer is:
spartan
'old greek' is the definition.
(I've seen this before)
'catches up with an' is the wordplay.
'catches' becomes 'traps' (trapping is a kind of catching).
'up' is a reversal indicator (in down clue: letters go upwards).
'with' says to put letters next to each other.
'traps' back-to-front is 'spart'.
'spart'+'an'='SPARTAN'
(Other definitions for spartan that I've seen before include "Of conditions, harsh and lacking comforts" , "Disciplined person" , "Frugal" , "Plain" , "Without comforts" .)