Is this property equivalent to Lusin's property (N) for continuous functions?
This does not work. What you call (N) like, is a rephrasing of Banach's condition (S): For every $\epsilon >0$, there exists $\delta>0$ so that $|F(E)|<\epsilon$ whenever $|E|<\delta$. (This trivially implies (N) like; use regularity for the converse.)
These conditions are discussed in detail in Saks' book. The basic theorem is: (S) $\iff$ (N) and (T1) (Saks, IX.7.4)
(T1) denotes the condition that $F^{-1}(\{ x\})$ is finite for almost all $x$. This condition need not hold for continuous $F$'s satisfying (N); this is discussed in Saks, Section IX.6.