Drop out of old American university (4)
I believe the answer is:
omit
'drop out' is the definition.
(I've seen this before)
'old american university' is the wordplay.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'american university' becomes 'mit' (I've seen this in other clues).
'o'+'mit'='OMIT'
'of' is the link.
(Other definitions for omit that I've seen before include "Exclude; forget to include" , "Leave out or neglect" , "Leave something out" , "Disregard" , "Leave out, do not include (4)" .)