Can old money be done for ten thousand yen?
The substance of this style is the state of not having to think about price. Setting a budget contradicts it from the start. There is still a way through.
Let me concede the contradiction first. Old money does not describe a shape of garment; it describes the visible residue of inherited wealth. Logos are absent because the position is already legible without them, and the idea of avoiding a cheap appearance does not arise inside it at all. By attempting it for ten thousand yen we are standing outside that. What can still be done is to reassemble what is visible from out here.
Where it gives itself away
The first tells in this style are cloth and shoes. Polyester catches light in a way you can read from across a street. Synthetic leather creases differently. Which also means: get the silhouette or the colour slightly wrong and the outfit survives, so long as those two hold. That is the answer to where the budget goes.
Allocation
Do not divide ten thousand evenly. I would put seven on shoes. Covering a top and a bottom with the remaining three is not realistic, so one of them has to come from what you already own. Many people will say that is not a ten-thousand-yen outfit. They are right. But I do not know a way to buy head to toe within that figure, and articles claiming otherwise are usually not counting the shoes.
Three colours, no more
Navy, undyed cream, dark brown. Finish inside those three and low unit prices hide surprisingly well. Palettes that read as cheap do exist; black with grey tends to expose coarse fabric. I cannot explain the mechanism well, but it holds in practice.
What not to do
Avoiding logos goes without saying. One more thing. Do not buy garments sold as old money style. Once the phrase is in the product name it is a word used by people describing the style from outside, not by people inside it. Buying by the name usually delivers something that only resembles the shape.
So
I think finishing the whole thing for ten thousand is not possible. But concentrating that ten thousand in the shoes and building the rest from what you have is a realistic way in, and there will be days when it reads perfectly well. In this style in particular, adding more pieces moves you further away.