@monoreposo
Now obviously the novel is more detailed, so they're glossing over some things by necessity here. No there are definitely big holes in his knowledge. Even the periodic table he wrote down is missing things. As is he'd probably fail High School Chemistry, let alone what he probably took (and forgot much of) in College. Also he isn't relying solely on his own knowledge, but delegating the responsibility unto others. He doesn't know how to do absolutely everything. For that there is taking time to think and figure it out, trial and error. Thus prior memory of everything isn't necessarily required.
Just a small example - I don't have a prior memory of the results of 165574929 x 2289354. I also can't do that in my head, but give me only a pen and paper and I'll solve it easily. You don't need to know all the answers from the start just as long as
you know how to solve the problems step-by-step. Thanks to his position, he has the time to think and solve problems. He also has people to help him do things, or devote their own time to solving the problems (delegation).
Designing a reasonable bridge is something I think any engineer could do (had they enough free time), although a bridge engineer would obviously be better at it. Hell if the requirements of the bridge where not too demanding, even I could do it (not sure how long mine would last) and I'm no engineer. Math... lots of math. An idea of the material strength and the forces it will need to deal with. Thermal expansion/contraction, wind, the weight it would need to carry, etc. It isn't easy, but it could be done. It could be done quite a lot better if you had prior knowledge/experience in the specific field of course. And even in that world, it isn't like he built the world's first bridge. There likely are even books about it in the nearby city he controls, though those probably don't mention steel frame bridges but should still be helpful as a starting point.
Lastly leadership - the previous prince was a useless person, yes. But he no doubt got taught lessons by very capable teachers. Maybe he didn't understand (or care) how to apply the lessons, but they are there in his memory. Plus I don't think it is odd to have a general idea of how societies function - especially as a modern person with access to mankind's collective knowledge at my finger tips. I do plenty of reading about various subjects just to kill time. And of course there is even general education where you're taught a variety of subjects - whether or not you're paying attention and committing them to memory is another matter. If you find absorbing new knowledge as interesting and rewarding (as I do), then it isn't strange at all. If you understand the general principal of a thing, you can use time and effort to figure out the finer details.
But back to leadership specifically, that too is something you can sit down, think about, and practice. In the beginning, Barov especially was doing all the work for him. I think they glossed over/skipped this too, but after waking up in that body, one of the early things others noticed was different was when he actually requested to go look over various reports. And at first he really did have a hard time wrapping his head around it all. He put in time and effort to get good at it... not a Gary Sue situation. But watching his go over reports daily isn't interesting, so the story isn't spending time on that. Teaching people things? I don't think he's good at it at all. Rather I think he's
terrible at it. He isn't making things relatable at all and as a result only a few people understand things the way he puts them. Luckily he has people better at it than him.
Perhaps the problems you have come from the abbreviated nature of this adaption, and in that case I'd suggest reading the novels.