Almost everyone is enthralled by the old university (4)
I believe the answer is:
yale
'university' is the definition.
(Yale university is an example)
'almost everyone is enthralled by the old' is the wordplay.
'almost' means to remove the last letter (most of the word but not all of it).
'everyone' becomes 'all' (I've seen this before).
'is enthralled by' means one lot of letters goes inside another ('enthral' can mean 'capture').
'the old' becomes 'ye'.
'all' with its last letter taken off is 'al'.
'al' going into 'ye' is 'YALE'.
(Other definitions for yale that I've seen before include "University in USA" , "University man" , "US Ivy League university" , "security device" , "Heraldic animal" .)