No, he did not. Caesar was in Asia Minor, fleeing from Sullan partisans, until 72, when he returned to Rome and was elected military tribune. While he might have served in one of the consular armies that Spartacus annihilated in 72, it's highly unlikely. None of our sources mention any military campaigns that Caesar participated in while military tribune, and none mention his participation in the Third Servile War at all. Military tribunes, particularly in the latter part of the Republic, often didn't actually serve in any military function at all--for most aspiring politicians it was a way to begin ascending the political ladder, and most either held a military (or other minor) tribunate or an aedileship (which Caesar held in 65) before entering into the senatorial magistracies (i.e. questor). If Caesar did actually encounter military engagements during his military tribunate, one wonders why such an important event, taking place during the monumental Third Servile War, would be overlooked when our sources dwell on the minor military campaigns of Caesar's time serving on Minucius Thermus' and Servilius Isauricus' staffs in Asia. In any case, Caesar could not possibly have served under Crassus, even if he did participate in the Third Servile War, which we have no reason to believe he did. Crassus did not enter the war until his praetorship in 71, when he was granted a special mandate to command eight legions, and by that time Caesar's magistracy had ended. Caesar did not hold another magistracy until his praetorship in 69, so he could not have legally been in the army unless he enrolled as a private soldier, which he of course did not.

Trying to fact check Kirk Douglas' Spartacus from the 1960s?