While Systema is a Russian martial art that can be tracked back to 948AD and has a long history of use, the modern form that you’re thinking of is only around a hundred years old. It was put together by the Communists in 1917, but only became available to the general public in the late 1980s. For reference, the earliest school founded in the United States began in 1993. Systema is a very effective killing form and, like all solid elite military training, was very tightly controlled.
It’s also not a form that is taught to children because of the necessary body postures and the maturity of the techniques that are being imparted. And due to the lack of an alternate sport form, there’s no reason to bring a child into it early because it won’t give them an edge. The only people who could teach Systema to this character would be a high ranking Russian ex-patriot, who also happened to be a family member and who wasn’t worried about getting caught either on foreign soil or in their own home country. This is unlikely given how tightly held the information about Systema was during the Soviet era. It would be too high risk for them to begin training the child, because it would end with them and the child being snatched off the street either by the local authorities or by foreign agents.
Also, when looking to develop a long term asset by beginning their training as a child, it’s important not to push them too hard before they are mentally ready for what they’ll be learning. A child doesn’t have the same concept of death or consequence that an adult does, you can break them by teaching them things that are outside their range of understanding. A villain won’t do it because if they are taking the time to raise a tool, they don’t want to have to go back and start all over again. Also, it’s a stupid decision and a villain who knows Systema will be clever enough to understand that. A heroic character won’t do it because it’s cruel and unusual.
You can raise them on a different martial art, like Sambo if we’re keeping with the Russian theme, and then move them onto Systema in their teens when they’re capable of grasping the psychology. Say around 14 to 15 years old. They’ll actually be much more effective long term going this route. Forget comic books and Cassandra Cain for a moment, the only reason someone puts a child into combat and asks them to kill is because they have a surplus supply and are looking to gain an advantage by causing the enemy to hesitate.
You can start training a child to fight as early as four years old (or even younger). But it stays with simple strikes like punches and kicks, simple movements like blocks and bringing it all together into a kata. You can do what they did in Ninjutsu, when fathers inducted in the ways taught their sons how to break into houses, how to go through someone’s things and then leave everything exactly as you found it, and even how to brew poisons by making it a game. That you can do and build them with the rest of the basic stuff they’ll need for later. But Systema has a focus on leverage, joint locks, and pressure points, a child under twelve just doesn’t have the level of muscular control to practice that safely.
I also meant what I said, modern Systema is on the upper end of combat arts. It’s a killing art that’s designed to not look like a killing art, it is beautiful to watch, and it is incredibly subtle with the way it links everything together. But it is doing a lot of very complex things which makes it difficult to write without a basic grasp of the combat psychology embedded in it, a functional grasp of anatomy, and a good sense of how killing styles are supposed to work (especially in comparison to Hollywood and normal sport combat).
Krav Maga or Sambo would actually have been easier choices with more readily available alternatives and probably some instruction on how to teach children. It’s more likely, however, that you’ll see students accepted in their early teens. The reason is that you don’t want to damage the developing tissues, tendons, and muscles in the joints too early because too much strain on an undeveloped body can lead to long term damage. There’s a reason beyond danger and a liability shield for why some martial styles come with a 18 and older price tag.
However, an older form of Systema is present in Russian folk dance. It’s not the form as it’s taught today, you may be able to partially reconstruct a combat form by having your character be raised on those. While most dance isn’t a good way to think about fighting, many of the older folk dances were ways that warriors celebrated and preserved their fighting styles when not in wartime.
So, think about it and check out the books and training videos by the leaders in the field. Try to remember also, that you’ll find Systema only in places with a high Russian immigrant population. The one thing it isn’t is common.