Does a non-EU citizen with Schengen visa have to provide proof of assets when crossing from the EU to Bulgaria, Cyprus or Romania?

Because you would be entering under short-stay rules, you must, among other things, have sufficient financial means to cover expenses during your trip and the return to your country of residence. Border guards can ask about if and if you fail to demonstrate that you fulfil the requirement, they can deny entry on that basis alone. It is a legal requirement and one of the standard reasons on the refusal of entry form.

But it does not mean that border guards ask each and every traveller crossing the border to produce bank statements or the like and it's certainly not required to have cold hard cash on your person. In most cases, you won't actually need to show anything and if they do ask, a credit card should be enough.

How thorough the border check is varies a bit but statistics show that in some countries it lasts something like 30 s on average. That's not a lot of time. So you can expect a couple of questions about the purpose of stay while the guard is looking up your name, visa and passport number in the relevant databases (to check whether you have been banned or flagged as dangerous, whether your passport has been reported as stolen or your visa cancelled). They would only move on to a more thorough evaluation (which might include questions about financial means) for a limited number of people.