a friend offered a cash reward for the answer
all i know is that he was a blues musician from the 1960s
I doubt this very much.
You have to turn a baseball cap around (or take it off) if you want an uninterrupted field of vision, particularly for things which are above your direct line of vision. Baseball players regularly turn their caps around for precisely this reason, and I would be astonished if it only occurred to them in the 1960s to do this. So I would guess that caps have been turned pretty much since they began to be worn during play.
It used to be common for goalkeepers in soccer games to wear a cap - a flat cap, not a baseball cap. I’ve seen plenty of old pictures of English games in which the goalkeepers cap is reversed, again I’m sure for this reason.
Caps would have been reversed off the field as well, either because people were engaged in activities which required a clear field of vision, or in imitation of players.
I suspect that this is pretty much as old as the cap itself.
I doubt anyone can claim to have first worn a hat backwards. Baseball catchers have been wearing their cap backwards since long before the 60’s.
Struggling very hard to take the OP seriously, I should point out that hats have been worn for nigh on 10,000 years. I’m pretty certain that someone in that time had occasion to put their hat on backwards.
perhaps not so much the act, but maybe when it became ‘cool’ was more what they were asking for.
Welders and cutters have worn caps with the bills backward for years before the '60s.The caps were worn this way to help prevent Red Hot Pieces of slag from going on the necks of the welders or cutters.
Photographers / cameramen?
The first person to put a partial brim cap on backwards was the genious that wore that cap on a windy day. After performing numerous mathematical calculations and subjecting that cap to rigorous scientific testing he was able to conclude that by turning the brim away from his face the cap was less likely to be blown off of his head. Subsequent tests supported the initial findings.
Offhand, I can’t remember the name of the genious. It might have been DiVinci, Newton, Pascal, Fermi or Einstein. It might have been someone else of similar stature. It might also have been a moron.
OK then, who first, while drunk, wore a lampshade backwards?
The lampshade doesnt matter. The reason? Well they where “Lit Up” at the time… so its a natural go together. 
Dang, NadaHappyCamper, you beat me to my first answer. I wear a pipefitter’s cap to protect the top of my head and wear it with the small bill pointing back becuase it gets in the way of my hood otherwise.
Rifles shooters are another category. How long have the marines worn a cap with a baseball style bill? You have to turn it around as it obscrures your line of sight in most positions.
OK, if we are looking for famous people with hats on awry, go with Jackie Coogan, in “The Kid” (1921) who wore his hat sideways.
I believe that one of the “Little Rascals” wore a baseball cap backwards, as well, but I can’t remember exactly which one.
Leo Gorcey, of the Bowery Boys wore a very oddly shaped fedora, which seemed to be backwards, although I am not sure it was.
I’ll try to think of some more.
Tris
Actually, Slip’s fedora just had the brim pinned up.
On the other hand Satch/Horace Debusey Jones (Huntz Hall) did wear a baseball cap that was often slewed around on his head in the same series.
Walter Matthau, playing the part of Oscar Madison, in The Odd Couple on Broadway in 1965, wore a backwards baseball cap. This may have popularized it in our modern time , along with the later movie and tv series.
Slightly off-topic, (or slightly on-topic depending on your point of view), I read an Ebert review a while back where he said that the first person to popularize the wearing of a baseball hat in everyday life was director Norm Jewison. I found that hard to believe, so I posted a thread about it in GQ. It was concluded that Ebert got the thumbs-down on that one. I sent him an Answer Man column but didn’t hear from him. (Although he did respond to another question I asked him about the use of puppeteers in Panic Room, so I forgive him).
You mean it’s “cool” to wear a baseball cap backwards?
Catchers turn it around because they have to, to fit the catcher’s mask on their faces. And catchers (like drummers in a band and archers in superhero groups) have “attitude.”
I don’t mean to brag, but I believe I was actually the first person to have ever worn a baseball cap backwards, sometime around 1984. Am I entitled to the cash reward?
According to Steven King (in Danse Macabre, his serious book on horror fiction in books and film), in the '50s monster movies there was always one wacky kid in the gang (often nicknamed “Wierdo” or “Scooter”) who wore his baseball cap backwards so you could tell how wacky he was…
If anybody would, I’d think Elvin Bishop would, but I can’t even find pictures from that era.