This has nothing to do with being a Lévy process or even with randomness.
Assume that the function $f:t\mapsto f(t)$ has no positive jump. Let $t_y=\inf\{t\gt0\mid f(t)\gt y\}$. Assume that $t_y$ is finite and that $y\gt f(0)$.
Then $f(t_y-h)\leqslant y$ for every $h\gt0$, by definition of $t_y$, hence $\limsup\limits_{s\to t,s\lt t}f(s)\leqslant y$. Since $f$ has no positive jumps, this implies that $f(t_y)\leqslant y$. On the other hand, if $f(t_y)\lt y$, then $f(t_y+h)\lt y$ for every $h\gt0$ small enough, otherwise $f$ would make a positive jump at time $t_y$. This is in contradiction with the definition of $t_y$ hence $f(t_y)=y$.
Likewise, $s_y=\inf\{t\gt0\mid f(t)=y\}$ yields $f(s_y)=y$, but $s_y\lt t_y$ is possible.
On the other hand, if $f$ has positive jumps, everything is possible, even that $f(t_y)\lt y$. If $f$ is càdlàg however, $f(t_y)\geqslant y$ but $f(t_y)\gt y$ may happen (consider the function entire part).