Castigate, scold and rebuke all mean to speak harshly to someone. However, 'castigate' is considered more severe.
Also, in spoken English, you would almost always hear 'scold'. The other two are used mostly in writing.
'Scold' can also be used as a noun to mean someone that rebukes often or constantly.
Pompous means self-important - giving an importance to one's self that is not realy earned.
Arrogant means that a person thinks they are better than other people.
While 'pompous' and 'arrogant' often go together, they do no necessarily mean the same thing.