I'm looking to travel between Kiev and Minsk next week by night train, however I'm not 100% sure on the dates. Is it reasonable to assume I'd be able to get a last-moment ticket or do Ukrainian trains often get sold out?
I've never went on this particular route, but as a policy you should book post-Soviet long-distance trains in advance.
People used to book them month in advance, but now two weeks is usually sufficient off peak season.
It usually goes like this: people buy out cheaper and best tickets first, leaving only more expensive or less convenient ones as time passes. In my experience you're unlikely to be stuck hard without a ticket but totally possible to over-spend two times on a ticket (you'll get better class on posh train you could totally live without).
A few days (3?) before train departure an influx of available tickets might occur, as reserved seats and seats on passing-through trains became available for booking. If you like to gamble this might be an option.
How last-moment is your last moment? Be sure to monitor ticket situation every few days.