Final answer:
The word archaic in this sentence means old-fashioned.
Step-by-step explanation:
In the sentence provided, the word archaic means old-fashioned. The word is used to describe the shirt collar, implying that it is a style that is no longer in fashion. This can also be inferred from the mention of the archaic Gladstonian shirt-collar, which further emphasizes the old-fashioned nature of the collar.