One at the Boar’s Head pub upset old Prince Hal, initially (8)
I believe the answer is:
bardolph
'one at the boar's head' is the definition.
(drunk in several Shakespeare plays)
'pub upset old prince hal initially' is the wordplay.
'pub' becomes 'bar' (I've seen this before).
'upset' is an anagram indicator.
'initially' indicates taking the first letters.
The first letters of 'prince hal' is 'ph'.
'old' anagrammed gives 'dol'.
'bar'+'dol'+'ph'='BARDOLPH'
(Other definitions for bardolph that I've seen before include "Falstaff's crony" , "Falstaff's corporal" , "'Knight of the Burning Lamp'" , "One of Falstaff's associates" .)