Probably the best option for such trips can be Slovak CityStar ticket (ZSSK-RZD).
Until December 2009 it will cost for two of you only just over 200 euro + sleeper supplements (Kosice-Kiev app. 20 euro per person, Kiev-Moscow depends on train (much cheaper in Ukrainian UZ carriages), Moscow-Vladivostok app. 220 euro per person in "kupe").
It means that cheapest train journey "any station in Slovakia"-Vladivostok and back together with all the sleeper reservations will cost app.700 euro for two of you. While regular domestic tariff one way ticket Moscow-Vladivostok for one person today costs app. 18500 RUB or 410 euro (train #2). I think the correct choice here is obvious
The only thing you need is to get from Budapest to any Slovak station and purchase there CityStar + sleeper reservations for the first leg of your journey. Probably faster and easier it will be to go from BP to Kosice, but Sturovo or Bratislava works as well