I think this was a hoax, probably for clickbait and it gained traction by being copied by other websites.

Russia is an IMF, World Bank, and BIS member state. I suspect that Russia would be asked/forced to leave one or more of those organizations if they were to replace their central bank.

Russia is a founding member of BRICS, which is an alternative IMF. It's going to take many years for them to develop BRICS but they are making progress. BRICS is a minnow compared to the IMF, WB and BIS who are all part of the corrupt banking cartel that controls the world.

Officially, the Central Bank of Russia (CBR) is already owned by the Russian Federation and Rothschild isn't mentioned.

You can view the Constitution of the Russian Federation here.

Following are all the sections from the constitution that talk about the CBR:

Article 75
The monetary unit in the Russian Federation shall be the rouble. Money issue shall be carried out exclusively by the Central Bank of the Russian Federation. Introduction and issue of other currencies in Russia shall not be allowed.
Protecting and ensuring the stability of the rouble shall be the major task of the Central Bank of the Russian Federation, which it shall fulfil independently of the other bodies of state authority.
The system of taxes paid to the federal budget and the general principles of taxation and dues in the Russian Federation shall be established by federal law.
State loans shall be issued according to the rules established by federal law and shall be floated on a voluntary basis.

Article 83
The President of the Russian Federation shall: ...
d) present to the State Duma a candidate for the appointment to the post of the Chairman of the Central Bank of the Russian Federation, raise before the State Duma the issue of dismissing the Chairman of the Central Bank of the Russian Federation;

Article 103
The jurisdiction of the State Duma includes: ...
d) appointment and dismissal of the Chairman of the Central Bank of the Russian Federation;

Unofficially, Rothschild may own the CBR but I'm yet to see any documentation to prove this. I'm sorry that I couldn't provide you with the proof that you were as I would have liked to see it myself but I hope this has helped you somewhat.

Let me know if you find anything and good luck.