Class-conscious one’s to be tearful, preserving name (4)
I believe the answer is:
snob
'class-conscious one's' is the definition.
(someone who's judgmental about social class)
'be tearful preserving name' is the wordplay.
'be tearful' becomes 'sob' (to sob is to cry tears).
'preserving' indicates putting letters inside.
'name' becomes 'n' (common abbreviation e.g. on forms).
'sob' enclosing 'n' is 'SNOB'.
'to' is the link.
(Other definitions for snob that I've seen before include "He thinks that he is socially superior to others" , "who puts on airs" , "One looks down on others" , "Nose-in-the-air person who thinks he is socially superior" , "One has pretensions" .)