This isn't something serious, just nationalism and/or cronyism. A real domestic processor project? It wouldn't be "dozens of millions of dollars" it would be tens of billions. Intel spent $10 billion on R&D... in 2013 alone. TSMC, who's just a fab not a designer, spent $1.4 billion in 2013.
Semiconductor manufacture is EXPENSIVE. A single modern fab easily tops a billion dollars to build, more like $3 billion. That's just to build it, running it and upgrading it can easily cost that much again over a few years. That is projected to grow to about $15 billion for a high end fab in 2020. All that, and you only have the ability to make chips, you don't actually have any chips to make.
Designing chips is again expensive. You need a bunch of smart, skilled, and experienced engineers and they need to put in a ton of work. It takes years. Companies that do fast design revisions have multiple teams that trade off working on chips, one team will be working on the next gen chip, another team on the gen after that, so that there is enough time to get the designs done.
So if Russia really wanted their own chips, like their own design, their own production, and all that, and wanted said chips to be on the same level as modern chips from Intel, IBM, etc, well they'd have to spend a ton of money, and a good amount of time.
This is, as you say, posturing. License an existing core design (made by Western nations), build an older technology fab, and produce some low end chips that aren't really that useful.