1)... is this a busy route, would we have a hart time getting tickets only 3-4 days in advance? it is summer time.. ?
Not extremely popular, so probably nothing to worry about, especially since end of July already will be after EURO 2012 football championship, so doubt there will be any problems with tickets*.
2) how much are they, about?
Direct international ticket Budapest-Kyiv is 42 euro + sleeper reservation 2/3 is 19 euro for Ukrainian UZ carriage. 2/3 means 2nd class and 3 bed compartment, but in reality for UZ carriage Budapest-Kyiv 2/3 is 2/2, a kind of a bonus - you pay for 3 bed compartment, but one bed is not on sale. In case you'll use Russian RZD carriages (Belgrade/Budapest-Moscow and Bar/Split-Moscow during summer) which go with the same 'Tisza' train, then 2/2 (meaning 2nd class and, as you want, double occupancy) is 29 euro
3) is the any way we can get a ticket, so its only 2 of us in the room?
Yes, as said above, the thing you need is double bed compartment sleeper reservation (indicated by '2/2' in case of RZD carriage or '2/3' in case of UZ) is what you need. First one is additional 29 euro per person or second one - 19 euro per person.
* even in case everything will be sold out (imho, chances are rather low), then you can always travel with a change, thus Budapest-Chop (17 euro for the direct international ticket) and then any of the multiple domestic Ukrainian trains from Chop to Kyiv (roughly 85-95 UAH for 'platskarnyi' carriage (6 berth open compartment) or 130-150 UAH 'kupe' (4 bed compartment). This option also might be a good choice in case you want to save some money, since as you can see, traveling with a change in Chop gives you total like twice less comparing total of international ticket and reservation Budapest-Kyiv
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