So the text goes, “name address address name address name address address (etc.)”, right?
How do you know what a name and what the address is?
Do they come in with some formatting applied? (even though it's the wrong formatting) — because then you could use Search & Replace. Search for 'blue' text and put your Name style in the Replace With formatting. Search for 'black' text and put your Address style in there.
Are there blank lines between successive names, for example, a single Hard return? Because then you can apply “Address” to everything, then use GREP to format all paragraphs-after-a-double hard return. (Ask & ye shall receive guidance on this.)
Okay, had to try first ;) This GREP Find will find and replace either the first paragraph in a story, or just those preceded by two hard returns (and no spaces on that blank line, please):
^((?<!r)|(?<=rr)).
Make sure the Replace With text field is empty, and your paragraph style is in the Replace With formatting field.