Like so many Bronze Age ideas, Plato's idea of a utopia doesn't really work so well in the modern world and beyond. The world has changed too much in the past 2500 years for it to work now, and even more changes will make this system even less viable.
The crux of the issue is that in Plato's day, basically everyone had the same life. The vast majority of people were focused merely on making sure that they had enough food to last the winter. His system worked then because it wasn't as large of an issue to have a few people (say 1% - it's been a while since I've read Plato, I can't remember if he went into detail on how many people would be needed to be Guardians and the warrior class) work for the good of the state in matters of diplomacy, rather than working for the good of the state by working the fields.
Now, however (and in the future, presumably), we aren't focused on survival. We have amounts of food unimaginable to someone in Plato's day - I have four pounds of almonds in my desk at work so that I can have a snack if I get hungry between meals. There are entirely different problems faced by today's world, and the world of the future.
And the problem is that someone raised to be a ruler likely wouldn't have any firsthand experience with many, if any, of those problems. (Note: using problems of today as concrete examples, as I won't pretend to know what the problems of the future will be.) If they're raised from birth to be a ruler, they wouldn't have student loans, wouldn't know what it's like to live paycheck to paycheck. Why would such a ruler care about fixing the economy when they've never known the pressure of trying to find a job in order to be able to pay rent? How can they effectively make laws regarding new technology when they own nothing and (presumably) have never used it (a modern analogue would be, how could they rule whether Apple should help the FBI unlock the San Bernardino shooter's iPhone when they've never owned an iPhone)?
Coupled with that is the problem of scale. For a Bronze Age city-state, a ruler would only have to know about a small area of land. Scaling this up to a galactic empire? Let's say it's a small empire; they "only" have to know about a hundred planets. How can they possibly know what's best for all of these worlds?
In order to fix this problem, you'd essentially need many layers of government. Have a set of rulers for each major city/area on a planet, who report to a set of rulers for the entire planet, who report to a set of rulers for a region of space (and the planets therein), who report to the "over-rulers" who rule the entire empire. Each set of Guardians has their own group of warriors, and any higher authority can call on them (eg the rulers of a region can call on the warriors of all planets in their region if they need to quell a revolt). Judicial appeals can be escalated to the higher levels, if so desired, similarly to the current US judicial system. Laws themselves work the opposite way - more local laws override empire law, except in certain situations (murder, rape, etc can't be legal, period, but something like drug use is down to the local administration), to ensure that the laws accurately represent the local culture, customs, etc. There would need to be rules in place dictating how the next set of rulers is chosen.
Honestly it's starting to look like a medieval society, but in space - kings rule counts rule dukes rule mayors rule citizens, and the higher levels don't really care what the lower levels are doing so long as they pay their taxes. Just with groups of people acting as the rulers, rather than individuals. And that hasn't worked out too well historically. And even with those changes, you don't fix the problem of the rulers being out of touch with the real world, with what the general population is experiencing.