Certainly if there were no ceiling functions around I would know what that inequality implies. I'm just not sure what else I have to be worried about.
My intuition tells me that I should just be able to conclude that $x<y$, however I'm concerned there's more to the ceiling function than meets the eye. I know that if the input is an integer the function will return itself, if it's not an integer it will round up to the nearest integer.
Edit: Certainly if $x<y$ then it's not necessarily true that $\lceil\frac{x}{3}\rceil < \lceil\frac{y}{3}\rceil$. For example take $x=2$ and $y=3$, obviously $x<y$ but $\lceil\frac{x}{3}\rceil = \lceil\frac{y}{3}\rceil$.
Your edit gets the implication backwards. It is true that if $\lceil \frac x3 \rceil \lt \lceil \frac y3 \rceil$ then $x \lt y$. It is not true that if $x \lt y$ then $\lceil \frac x3 \rceil \lt \lceil \frac y3 \rceil$, which your example shows. You can add $x \lt y-1$ to the conclusion when $x \equiv 0 \pmod 3$ or $y \equiv 1 \pmod 3$