I don't think I can do the full question justice, but I can give you a minor, albeit somewhat odd, angle on it: Falconry.

Frederick loved falconry. He loved collecting the birds themselves, he loved exotic and rare raptor species, he loved flying them at game, he loved arguing with his staff of falconers over the best training methods, and he loved competing with his court members to see who could take the most quarry in a single day. I assume the smartest among them made a habit of losing to him regularly.

He wasn't alone in this. This was an era in which falconry, and hunting more generally, played an outsized role in the lives of many royals and aristocrats, to the point of having its own little impact on history. There's a semi-funny story from the siege of Acre in 1191 in which the French king, Philip Augustus, lost an incredibly valuable white-morph gyrfalcon while hunting a few miles away from the city. (He was likely after the houbara bustard, which is the traditional quarry for falcons in the region. Sadly, the bustard's modern population on the Arabian peninsula has been almost completely wiped out by Saudi royals who go hawking with Land Rovers and dozens of falcons at a time.) The bird failed to make a kill, flew over Acre's walls and into enemy territory, and was captured and brought to Saladin. Once Philip figured out where the bird had gone, he sent emissaries to negotiate for its return. Saladin was having none of it, and Philip was reduced to begging and offering a small fortune to try to get it back. This is napkin math and shaky at best, but I think he offered the modern equivalent of $180,000-220,000. Saladin decided to hold onto the bird because: a). gyrfalcons, particularly white-morph gyrfalcons, were huge status symbols: b). it was actually a lowball number for the value of the bird, and c). it was a convenient little fuck-you to the guy who was besieging the city.

This wasn't the last time that birds of prey played an unwitting role in the region's oft-fractious politics. In 1396, John the Fearless (son of Philip the Bold, then duke of Burgundy) got himself and twenty four of his knights captured by the Turks either during or immediately after the Battle of Nicopolis. The Turks under Bayezid I refused to give them up until negotiators settled on the eye-popping sum of 200,000 Venetian ducats (napkin math gets shakier here but this could easily be hundreds of millions of dollars) and 12 gyrfalcons. They wanted the birds more than the money; because the gyrfalcon is indigenous only to high northern latitudes, people in or around the Middle East didn't have easy access to them.

Anyway! The difference between Frederick and most of his noble cohorts was an intense intellectual curiosity -- and of course, the means to satisfy it. Being as familiar with the sport as he was, he couldn't help but notice that many of the existing scientific treatises on birds of prey, even those by the venerated Aristotle, either didn't match what he saw in the field or were woefully incomplete. He hired scholars to translate Greek and Arabic texts on birds for him. He started dissecting dead hawks and taking extensive notes on their anatomy; a significant portion of his resulting book is about nothing but bird organs and their function. He observed the different species' behavior, both in the wild and captivity, critiqued different training techniques, wrote extensively on the use of gear and strategy in different environments in the field, and wrote detailed descriptions of the various species used in falconry and their traditional prey.

The end result was an enormous book titled De arte venandi cum avibus, or "On the Art of Hunting with Birds." It is arguably the first modern manual on the sport of falconry, which I know is a very strange thing to say about something that was written in the mid-13th century, but a lot of the practices, observations, and suggestions made in De Arte are instantly familiar to ornithologists and falconers today. With the exception of satellite tracking for the birds, the growing use of North/South American species, artificial insemination, and the introduction of Aylmeri jesses in the 20th century, falconry really hasn't changed much since antiquity. What has changed is our scientific and factual understandings of the birds and their behavior, and Frederick made an unusually early contribution to this.

Here's an excerpt. I had to borrow this from the very few pages available on Google Books; my copy of the English translation has apparently disappeared into a black hole somewhere. Anyway, this is from his chapter on migratory patterns:

We notice also that when a favoring wind springs up, whether by day or night, migrating birds generally hasten to take advantage of it and even neglect food and sleep for this important purpose. We have observed that migrating birds of prey, that have begun to devour food we have thrown to them, will abandon it to fly off if a favorable wind begins to blow. They would rather endure hunger and travel day and night than forego such an advantageous opportunity.

The calls of migrating cranes, herons, geese, and ducks may be recognized flying overhead even during the night, and not, as Aristotle claims, as a part of their efforts in flight; they are the call notes of one or more birds talking to their fellows. For example, they understand wind and weather so thoroughly that they know when meteorologic conditions are favorable and are likely to remain so long enough to enable them to reach their intended haven.

This was centuries before the scientific method was a thing, but there's a distinctly modern feel to how Frederick researched and wrote the book. Not only did he spend a lot of time watching birds (and likely assigned underlings to do it for him and report back), but he also experimented to see if their behavior varied in the presence of food or other incentives. Interestingly, his correction of Aristotle here is also the modern ornithological understanding of migrating bird calls, e.g., geese honk to communicate with each other, keeping the flock in formation at a consistent speed.

I'm hesitant to address why Frederick might be considered the "first modern ruler" in a political sense, but you can make a strong argument that his commitment to empirical observation and questioning of scientific orthodoxy in De Arte certainly set him apart from most of his contemporaries. However, you can also argue that Frederick's work reflects the huge push in the academic establishment of the 13th century toward logical thought and rigorous observation; he overlaps with such luminaries as Albertus Magnus, Grosseteste, and Roger Bacon.

So was Frederick ahead of his time, or more a reflection of its best sensibilities?