Estimate can be either a noun or a verb.  In both instances it is related to making an educated guess about the quantity or value of something, either the act of making the guess (verb) or the guess itself (noun).

A synonym is a word that means close to the same thing as another word.  So synonyms for estimate the verb would include appraise, assess, valuate, gauge, and guess.  Synonyms for estimate the noun would include appraisal, assessment, valuation, and guess.

An antonym is a word that has the opposite meaning of another word.  This is a little trickier since you could go in two different directions.  If you wanted to mean the opposite of making an assessment, then an antonym would be disregard.  If you wanted to mean the opposite of making an educated guess in the sense of making a precise calculation, then an antonym would be calculate.