Objects, you can refer to as 古い with impunity. But I have never once heard a person referred to as 古い, so for a long time I thought the word just didn't apply to people. Still, searching 古い男 on Google gets pictures of old men. I imagine it's probably best to refer to old people as something like おじいさん or 老人 if their age is relevant, though.
To us native speakers, 「古{ふる}い男{おとこ}」 does not mean "an old man". This is why direct translation could be dangerous at times.
「古い男」 to us generally means "an old-fashioned man". In other words, it is synonymous to 「古いタイプの男」、「価値観{かちかん}の古い男」、「考{かんが}え方{かた}の古い男」, etc.
Therefore, 「古い男」 does not necessarily have to be biologically old because it is his personal values that matter here.
To refer to "an old man", we use 「年配{ねんぱい}の男の人」、「老人{ろうじん}」、「年老{としお}いた男性{だんせい}」、「おじいさん」, etc. We do not use 「男」 by itself nearly as often as you use "man" in English.
Finally, it might help to remember that 「古い」 is the antonym of 「新{あたら}しい」 ("new") and not of 「若{わか}い」 ("young").
But I have never once heard a person referred to as 古い,
We use 古い also 新しい for a person not for describing his/her age but for describing his/her way/tendency of thinking.
As you know 古い is an adjective, and an adjective has two different ways of use as a predicative use and an attributive one.
An attributive use of 古い is already answered using the example with 古い男, so I'll answer 古い in a predicative use.
古い in 彼は古い means 彼の考え方は古い His way of thinking is obsolete/old-fashioned, while 彼は新しい means His way of thinking is novel/original/innovative/unconventional/audaciou/avant-garde.
Both ways of use of 古い could be used, but I think 彼は古い/新しい is used more than 彼は古い男/新しい男です.
EDIT
A good way to refer to "an old man" is お年寄｛としよ｝り like （車内{しゃない}では、）お年寄りに席｛せき｝を譲｛ゆず｝る to yield one's seat to an elderly person (on the train).
