In this sentence the word "overall" would mean all parts. So all parts of the school system at this university come together to make it the best in the country. I think the sentence would sound better with the word "the" in front of best, but it is right both ways.

You are right, that the sentence should have the word "the" in front of the word "before". The correct sentence should look like this: "It has some of the finest universities and probably best overall school system in the country." The word "overall" means all things considered. For example, I could say "the corvette is faster, but the porsche is a better car overall" to mean that if you look at all of the features together, the porsche is better.