I'd argue the way they build light general aviation aircraft today: by using mundane tools and mundane materials and mundane skills plus a lot of enthusiasm. It's just that "mundane" would be defined in 2050s-2060s terms.
The ability to build an aircraft in one's garage today would be nothing short of magical for somebody from the 1880s, or even the 1950s when certain tools and techniques were prohibitively expensive and complex for the layman to use. But tools and techniques have been advancing, and one finds high technology off the shelf when mere decades prior it would have been locked in the vaults of big corporations or even national intelligence agencies.
I know out of a missile, but they put in a working matter-antimatter engine, with arm extensions(which I assume works by hydraulics).
Just to tackle a detail, the arm extensions might be more "space-proof" if they worked on magnetics (i.e. with fancy electric motors, probably of a lubrication-free variant). Hydraulic fluids and the extreme temperatures of space might not mix well.
How did they even make a Anti-matter/matter engine?
By applying 2050s tools and techniques, I guess. For all we know, matter/antimatter annihilation powerplants were fairly standard military hardware in the Trek 21st century, and Cochrane's project could make use of not-quite-top-of-the-line military surplus. Or then m/am reactors were already common in civilian life as well, and could simply be bought, even though they sound like high-risk, low-gain technologies to me and probably thus not of interest to the commercial sector.
How did they accomplish this after a nuclear 3rd world war, with more than half of the population gone?
The war might actually have helped a bit.
After WWI, there was lots and lots of surplus aviation technology just lying about, and anybody could buy or scavenge it and build record-breaking aircraft - and many did. After WWII, there was even more. It wouldn't surprise me a bit if after WWIII, every halfway decent military town (with a garrison, an airbase or an industrial plant) was littered with surplus spacecraft.
Really, when we look at ST:FC and see a rocket, we shouldn't be going "Gosh and wow, a rocket!". We should be going "Poor sod, having to use rockets to go to space - what is this, the 1980s?". In the Trek universe, far more advanced spacecraft were being
retired in the 1990s already. Cochrane's little home rocket is probably best compared to a sand buggy built out of a VW Beetle chassis, while the posh neighbor is building a small jet aircraft out of a commercial kit.
Timo Saloniemi