Using 'mine' before or after a noun instead of the more modern 'my' is indeed archaic/historical. Prior to Shakespeare's time, 'mine' and 'thine' became 'my' and 'thy' before words starting with a consonant (like 'an' and 'a' today).
By the time the Bard was writing, the language was in transition. He uses 'mine' before 'host' (the 'h' was silent, this suggests) but elsewhere he uses the modern practice, e.g. 'But thy eternal summer shall not fade' (Sonnet 18). There, 'thy' is stressed perhaps rendering the liaison unnecessary. Compare 'Dear heart, forbear to glance thine eye aside' (Sonnet 139), where the stress falls upon 'eye'.
Up to the 19th century or slightly later it was sometimes used in poetry and song ("(Oh Baby Mine) I Get So Lonely" is a 1953 American song). Apart from that, it is widely used in modern British English, in 'mine host' (even when pronouncing the 'h') as an arch or jocular way of referring to the person who owns a pub (public house, bar). Other jocular usages are possible, e.g. Oh brother mine! (if you were exasperated with him, perhaps). Using it in other situations, e.g. mine car, mine girlfriend, mine house, etc, in everyday conversation or writing, while it would probably be understood, would be regarded as silly or mistaken.
humorous The landlord or landlady of a pub.
‘mine host raised his glass of whisky’
Host (Oxford Dictionaries)