Bound to be tired right out? (4)
I believe the answer is:
tied
'bound' is the definition.
('tie' can be a synonym of 'bind')
'tired right out?' is the wordplay.
'right' becomes 'r' (common abbreviation).
'out?' suggests deleting specific letters.
'tired' with 'r' taken out is 'TIED'.
'to be' is the link.
(Other definitions for tied that I've seen before include "Restricted, bound" , "(Of pub) owned and run by a brewery" , "Bound; placed equal" , "United" , "Bound to diet" .)