Since residue classes modulo $p$ are not numbers but sets of numbers with several operations defined on them, i don't think we can compare them at all. Yet i'm not sure.
For sake of an example lets take $p = 2$. We can claim that,
$$[0]_2 < [1]_2.$$
But wouldn't that mean any odd number is greater than any even number? Which is obviously false...
After all, if we generalize this question, does an ordered finite field exist?