I don't have an answer for you, but you might find an answer to at least the first part of the title, if you go look at even older languages that modern Icelandic is derived from, and still resemble quite closely. It's been a while since I read old Norse (and pre-Norse) language, but the use of apostrophes does seem familiar when in the Latin scripture.

The oldest known printed apostrophe in Europe is from 1496, and the language of that book (De Ætna) was indeed Latin - soon to be followed by Italian, French, and so on. Your comment helped to remind me that the pressure to standardize punctuation, in any language, until the printing press came about, was rather limited, and that various conventions to indicate omission or clitics must have been springing into scribal use constantly, even without being called "apostrophes" at the time. Still, I'm hoping that as the OP points to a newspaper text from early 20th century, there must be something black and white to say on printed Modern Icelandic, too.