The Bud was at 30 John Street. No, there was never a tunnel from Holyoke City Hall to the Bud. Perhaps city officials would love to have such a tunnel to got to their favorite pub but no evidence is there.
Sanborn map analysis:
Sanborn 1884 map shows that it is a bottler of alcohol and that is wooden and 2 floors
Sanborn 1889 map same
Sanborn 1895 map shows it as a candy factory and it is 3 floors and brick
Sanborn 1915 map it is now for wholesale alcohol
Sanborn 1956 map shows that it is a restaurant