Sierpinski showed in Hypothèse du Continu that there exist a set $A \subseteq \mathbb{R}$ with the cardinality of the continuum whose image under any continuous function is a nullset. (Proposition $C_3$ on page 49. It is a consequence of Theorem 2 on page 41 and - assuming CH - the existence of a Lusin set).
He also showed that there exist a set $A \subseteq \mathbb{R}$ with the cardinality of the continuum whose image under any continuous function is perfectly meager. (Propostion $C_{30}$ on page 86. It is a consequence of Theorem 1 on page 85 and - assuming CH - the existence of a Sierpinski set).
Does there exist a set $A \subseteq \mathbb{R}$ with the cardinality of the continuum whose image under any continuous function is a perfectly meager nullset ?
You are misquoting Sierpinski. Although one can construct an uncountable set of reals which is both perfectly meager and universally null in ZFC alone, the existence of a size continuum perfectly meager or universally null set is independent of ZFC.