Yes. That is correct. You could also say, "He proudly boasted about his success." English is a funny language.. :P

We don't say 'boasted proudly' it is not correct. He boasted about his success. He announced his success proudly. You don't use them together because if somebody is boasting, we already know they are proud of their accomplishment. But boasting and being proud are two different things as well. Boasting is usually an arrogant act where pride isn't.

I think it should be "He boaster proudly about his success." if you want your word order that way