The simplest option is two overnight through trains, Milano - Wien one
night and Wien - Bucuresti a following night. Both are available using
"global pricing" (which means a single fare including on board
accommodation), though I suspect that won't meet the OP's request for
the cheapest solution

Sorry I wasn't clear. Indeed, I don't want to spend a LOT but there are
other factors than price. Even the bus is not out of the question, nor
the flight. But the train, with a tourist stop along the way is an
experience that appeals to me. Night train would save money, but
eyeballing the country along the way is worth the cost of a night's
lodging (under €25 is pretty easy to find).

I've done the sleeper train from Budapest most of the way to
Bucuresti - problem is, it crosses the border at about 2am and
it'll only just be getting light when you hit the interesting
bits of Transylvania. I'm not sure what alternatives there are.

I have been from Budapest to Rijeka by train - easy to get to
Rijeka from Venice. That takes you through some attractive
countryside in Slovenia.

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