Is the visa-free stay of a Ukrainian citizen, which ends in a period of an epidemic emergency, extended automatically like a visa (until the 30th day after the lifting of the epidemic emergency)?

YES. The regulations introduced on 18 April 2020 make it possible to extend the legal stay for foreigners staying in Poland based on of visa-free travel, Schengen visas and residence documents issued by the other Schengen States until the 30th day following the date of the cancellation of an epidemic state or epidemic emergency (the one that will be last in force).