The second sentence is wrong, or at least incomplete.
The first sentence translates to something like "She already turned three (years old) last month". The expression of time can be placed in German like you see in the sentence. You might say, the sentence divides up this way
Sie | ist | letzten Monat | schon | drei (Jahre alt) | geworden.
The second sentence would verbatim translate to something like "She already turned three month last". As you see, the age would need to be in plural ("drei Monate" instead of "drei Monat"). Also, the expression of time is incomplete (last what?)
A complete sentence might look like this: "Sie ist letzten Dienstag schon drei Monate geworden", "She already turned three months last Tuesday".
As you see, both the first sentence and the (completed) second one have significantly different meanings.