The connecting train from Khabarovsk to Nakhodka was effectively a "boat train", and only ran on the days the ship was sailing to Japan, which was about every 3 days in summer, cutting back to just once every 10 days in winter. The ship took two nights to get to Yokohama, there were three ships in service in the summer, just one in the winter.
Nakhodka is less than 50 miles east of Vladivostok, but the connecting train ran for about 500 miles from Khabarovsk, sharing the Trans-Siberian route almost all the way to Vladivostok and then turning off shortly before reaching there. The Nakhodka line follows a decidedly sinuous course through the deeply forested coastal mountains; it was built by the Soviets in the 1930s along with the port installations at its end, as a secondary transhipment point to Vladivostok.
Foreign passengers were assembled at Khabarovsk, the Nakhodka train didn't connect as such directly with the daily "Rossiya" service from Moscow, but departed in the late afternoon for an overnight run, whereas the Rossiya arrived there late evening, forcing through passengers to stay in the Intourist hotel there. This was actually a benefit because after 7 days running from Moscow, although on-time punctuality was notably good, on the occasions when the Rossiya was late it was sometimes VERY late.
It is likely the cars used were among the better SZhD (Soviet Railways) stock, as indeed they were on the Rossiya itself; although much of the passenger stock looked identical from the outside, the interior furnishings varied considerably (as is still the case on Russian Railways today). I believe the Nakhodka train was Soft Class only. The Soviet Railways gradually built up their assets over time, and by the 1980s had actually got their standard vehicles to a pretty good standard (toilets excepted, alas), and certainly spacious, and they then rolled out very considerable quantities of stock, much of which is still the bulk of the Russian Railways vehicles today. The standard vehicles, with two or four berth sleeping compartments, were built in the Soviet Union, but many of the specialist restaurant cars etc were built in the German Democratic Republic, while all of the Soviet passenger train electric locomotives were built, you may be surprised to learn, by Skoda in Czechoslovakia.
In passing, the railways (and the country, and the general expression internally for their country) in those times was the Soviet Union, not Russia, although the name of the Moscow-Vladivostok train is a bit of an exception to this; it did of course travel entirely within the Russian Republic.