Is it possible to create a style such that the first line of the style
has one character formatting, but subsequent lines have a different
character formatting?

The case I am considering is a multiline title header, where only the
bottom line of the header should be underlined (but the whole thing
should be bold). The header is not broken up with paragraph marks, but
rather by wordwrapping or hard-returns line breaks.

This sort of formatting is common in legal documents.