Not even close, the party will start tomorrow. The party will begin tomorrow, for me it's either one. Actuall we would say before the start of the pearty, let the party begin. A more appropriate use of the word.
Given how similar Present Tense and Future Simple may be in Polish, we are very strict in terms of using tenses in the translation. "zacznie się" is Future Simple, therefore we require "will begin/is going to begin". Your sentence is "zaczyna się".