29 April 1975..Operation Frequent Wind

Equation said:

How the heck was he able to land a Cessna on a carrier without arrested cable and hook?

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Quite easily. Light aircraft like the one shown are already in military terms, STOL (Short Take Off and Landing) aircraft and the deck of a carrier is a lot longer than they would require for a runway on land. Back in the 1960s British Army Air Corp light observation aircraft regularly operated from the decks of the Commando Carriers (LPH) Albion and Bulwark which lost their catapults and arrestor wires when converted:

Yup. The whole reason for the wires and gear on a carrier deck is the heavier weights of armed aircraft higher speed military fighters. This lead to the cats and traps. The jet age made these issues even worse due to there higher mass and speed demanding two additions to the carrier the angled deck and the meatball.
Fighters in the second world war used cats mostly to open up deck space to launch of more aircraft same for traps to allow more space ahead to allow for aircraft to take off and land at the same time. But as oriven in the launch of the Doolittle raid a fairly large twin engine aircraft can take off the deck of a Yorktown class carrier. And later a C130 conducted multiple units assisted take offs and landing on the USS Foristal.
Midway though not as large as a Foristal class the first of the super carriers, is larger than the Essex or Yorktown class and with an angled deck she has a substantial landing way for a slow and light weight single engine puddle jumper. Heck the pilot could probably have even taken off again without the cat.

Light aircraft like the aforementioned Cessna can take off from a carrier in less distance than the length of the catapult track, assuming the ship is steaming into wind! Cats and traps are for heavy and fast aircraft; light and slow can do it without much help. 1950s ASW aircraft like the Tracker and the Gannet could still perform free takeoffs and would do so if the deck wasn't too crowded. The Brazilian carrier Minas Gerais only operated Tackers and helos, the former frequently used free takeoffs as late as the 90s when her catapult became non operational for a few years.

Yeah but this guy did it without practice ever landing his Cessna onto a carrier. So that meant he had to get it right the first time or else he swimming with the fishes.

No not really, as Obi Wan stated these military Cessnas are STOL birds. There slow speed and light weight means that they can land on very short feilds. His approach speeds would have been very slow is weight is low so he doesn't have the same momentum as a F4 fighter.
So his landing is far far easier he lowers his speed to just the edge of a stall and touches down on the deck and he's basically down. When I was a kid my grandfather who was a Local Shariff deputy used to take me to the office at the local airport, not a big regional or hub but a little short feild where birds like these took off all the time they use a little grass feild point into the wind throtal up and take off in a few yards. Landing was a quick tap on the ground and a very short roll most of it being just taxi roll.

I launched off the JFK in October 1972 in a C-1 Trader COD. We deck launched from the angle deck. No catapult.