Translation:Kateřina finished school one year before me.
If it was written "...one year before I did." then yes, it would be grammatically correct. The reason why is that the "did" completes the sentence by joining with the "I" to form a complete clause. However, without the "did", using the nominative form "I" is grammatically incorrect.
"Before" is a preposition and demands an object (more specifically, it demands the accusative form of "I", which in English is "me"). Thus, it is grammatically correct to say "...one year before me."
Here are some additional points to consider:
The word "before" can be an adverb, a preposition, or a conjunction: (a) I had never seen him before vs. (b) I stood before the king vs. (c) He got up before I did.
When "before" is a preposition, it is followed by a noun in the objective case. In contrast, when "before" is a conjunction, it is follwed by a clause, the subject of which is in the nominative. So, "He stood before me" but "He got up before I did".
Now we get to the interesting part. The verb in a clause following a conjunction can be hidden/suppressed. That is, instead of "He is as tall as I am", once can correctly say "He is as tall as I". (The word "as" is a conjunction here.)
So, on the one hand, it could theoretically be correct to say "K. finished school one year before I" if "before" is a conjunction in that sentence. But is it, or is it a preposition? There are arguments on both sides.
On the other hand, few nowadays would say, or even write, "K. finished school one year before I". Most would say "before I did", to make clear that "before" is being construed as a conjunction.
Compare "He is taller than I am" vs. "He is taller than I" vs. "He is taller than me". Do you consider "than" a preposition or a conjunction? There are arguments on both sides.
In any case, in Czech, the word "před" is a preposition followed by the intrumental case.