There is relatively scant information in English about construction costs in Russia and China. Frustratingly, even Metro Report, which does have some information about China, has only a handful of Russian examples with their costs stated; from perusing the articles Wikipedia links to, even Russian originals rarely state the costs of subway extensions.
Fortunately, Metro Report does have an article mentioning general costs. Be warned: the costs quoted below are somewhat higher than the specific figures I’ve found for individual projects.
Tunnels, including stations and depots, cost an average of 10bn to 15bn roubles per route-km to build, with construction of an extension lasting five to six years. Cut-and-cover methods can save 2bn to 5bn roubles and up to three to four years. Additional savings could be made by using double-track bored tunnels, which first appeared in 2014-15 in St Petersburg, along with top-down station construction. At some stations in Moscow, platform arrangements are being introduced with a platform on each side of a single track so that boarding and alighting passengers do not use the same platform; this leads to a 15% to 30% saving on the overall construction cost.
The PPP conversion rate is about US$1 = 24 rubles as of 2016-7. So the overall cost quoted is supposedly around $400-600 million per km, which is very high for a European country, and overlaps the American range (though the $500 million/km American subways tend not to be in city centers). In practice, the two specific lines cited in the article are cheaper, at $310 million/km for the Line 3 extension in Saint Petersburg (which is partly underwater) and $185 million/km for the Line 2 extension in Nizhny Novgorod; but both extensions have wide stop spacing even by Russian standards, and deep underground, stations dominate construction costs.
Look more carefully at the quoted paragraph. Using side platforms rather than island platforms is stated to reduce costs by 15-30% – presumably overall costs, not just station costs. This is because the caverns are simpler, especially if the stations are built cut-and-cover. Cut-and-cover overall is supposed to save 20-30% of the cost, taking the 10-15 billion figure as correct and not the lower figures of the Saint Petersburg and Nizhny Novgorod lines mentioned in the piece. If the lower figures are right, the saving is around half the cost, making cut-and-cover cost about the same as above-ground construction (an above-ground Line 1 extension is projected to cost $130 million/km).
I saw a different source, in French, make the same claim that cut-and-cover is about as expensive as elevated construction; I can’t find the reference anymore, but interested readers can Google “ciel couvert” and see if they can find the article. This was very much not the case in 1900-4, when New York was spending (in today’s money) around $39 million/km on the subway’s underground portions and $9 million/km on its elevated portions, but then again New York built els to be cheap and noisy, and it’s plausible that quieter concrete structures would cost more.
Another plausible explanation is that cut-and-cover has gotten relatively cheaper over time due to mechanization of street digging. New York and Paris built their subways with hand tools in the 1900s. Deep boring is more mechanized, but was already somewhat mechanized at the turn of the century, so it’s not surprising if the cost trajectory in the last 120 years has been more favorable to cut-and-cover. As it is, London’s early Tube lines didn’t cost more than the cut-and-cover lines of New York or Paris, nor did they cost more in the 1930s; the cost differential is thus a recent phenomenon.
Finally, on a more political point, it’s worth comparing Russia with other countries that used to be in the Soviet bloc, since they have broadly comparable incomes today and learned to build subways from the same place (i.e. the Moscow Metro and the Soviet triangle). Overall, Russian costs seem somewhat higher than in the rest of Eastern Europe: comparable to costs in Poland or a little higher, somewhat lower than Hungary (M4 was around $500 million/km), much higher than Bulgaria and Romania. Does EU membership and the package of reforms required for accession mean lower construction costs? It’s not guaranteed, but it looks like the parts of former communist Europe that joined the EU are doing better. Upper middle-income wages with good institutions can produce good results, just as the never-communist parts of Europe with comparable incomes, like Greece and Italy, have pretty low costs.