I saw this episode too and was concerned by that point. Brill's New Pauly says it's unknown whether he served in the war against Spartacus in any capacity. I double-checked Goldsworthy for you, and checked Meier, and neither of those biographers speculate about Caesar's military service in that campaign.

The speculation must be grounded in the fact that by 73 he returned to Italy from studying in Asia among other things, and got elected to the position of military tribune. Our information on that point is vague (Goldsworthy doesn't know if it was for 72 or 71), and the writers of the Netflix show must be guessing at which campaign he served in. The Spartacus campaign is attractive because of Caesar's later relationship with Crassus and Pompey - it would be a very tidy story if all three contributed to putting down Spartacus and that experience formed a basis for their later partnership.

It's speculation though.