WARNING THIS IS TOTALLY A JOKE NONE OF THIS IS REAL. READ MOD NOTE HERE

This was something I learned about during my time as an underwater archaeology minor while pursuing my degree. Fedoras (at least the ones we've seen worn for the past 100 years or so) actually have a long history and were only recently "re-discovered" about 170 years ago.

Prior to their rediscovery in early 1844 by a British Archaeological team digging in what used to be an Etruscan village, referred to as Turelbi in some older texts (though I haven't been able to find it referenced any later than 1918) in what today is Northeastern Italy and discovered a cache of preserved fur-pelted hats that were similar in design to what we would recognize a Fedora to look like today except it had two distinct corners at either end.

Anyhow, the team brought these items back to England to show off in an exhibit at the British Museum that ran for a few years. Oddly enough, years after the exhibit had closed, Fedoras began to become fashionable piece of head wear among men in England and France after a play written in 1882's main character wore a hat inspired in design by the fur pelt hats found at Turelbi and displayed at the British Museum. Fedora's then became the fashionable head wear of the bureaucratic middle class.

The original hats that were discovered have been lost unfortunately, supposedly they went missing while in transit between a museum warehouse and Liverpool (the hats were to be shipped to the Americas to be exhibited in recently opened American Museum of Natural History in NYC in the mid 1870's).

Source: Address to the Ethnological Society of London Delivered at the Anniversary, 25th May 1844

I have a followup question: some Classical sculptures, such as this bust of the Emperor Nero feature what we today might call a 'neckbeard'. Was this a widespread style of facial hair, and what kind of social demographic would it have been popular amongst?

Your friend is not entirely correct. A straw fedora would only ever be appropriate with, say, a white linen suit in the Caribbean, South America or a similar location, but keep in mind that the fedora was the go-to hat for decades, from the Prohibition to the late 1950s and early 1960s.

Humphrey Bogart wore them extensively.

Frank Sinatra frequently wore them throughout his entire career. Here we can see both Frankie and Dean Martin in fedoras, with Sammy Davis Jr. rounding it all off with a bowler hat.

Don Draper, portrayed by Jon Hamm in Mad Men, is another proponent of the fedora.

Fedoras, paired with a nice suit, were once the pinnacle of style, but that was half a decade ago. Wearing a fedora today takes a certain kind of audacity, even more so given how it has become identified with neckbeards. Celebrities like Johnny Depp have occasionally been seen wearing fedoras, but they get away with it because they're celebrities and are exceptionally handsome. Johnny Depp could wear a bagel on his head and still be more attractive than the vast majority of all men in the world.

So no. Forget about your fedora. It doesn't work in this day and age, and especially not a straw fedora.