Caesar almost certainly did not participate in the war, as it was all but over by the time he returned to Italy. Sulla died in 78 but Caesar didn't return to the city until 72 to stand for the military tribunate. Although technically speaking the Spartacus war was not yet over during his military tribunate it was pretty much wrapped up and he appears to have remained in the city, probably to canvass for the quaestorship--military tribunes frequently never actually saw a battle during their tenure. It's not technically impossible that he was military tribune in one of Crassus' legions (although at this stage in his career he had little connection to Crassus yet), but literally none of our sources mention it, they all mention instead his service on the staff of Thermus and against Mithridates. One wonders why, had he seen service against Spartacus, these relatively minor campaigns are mentioned in our sources over what must have been a vastly more important campaign against Spartacus, and (especially if he served under Crassus) a campaign which would've in all likelihood been quite formative.