S-Ranking a Mission requires 90% or better in each of the categories at the end of each match within a story section. These are usually % Health left, Continues used, % QTE complete. Bonus objectives and time spent have no bearing.
Health must be 90% or better (you will kick yourself the 3rd time in a row you finish on 89%)
Continues are self explanatory, don't fail the fight for a S-rank.
QTE are the various scenes that play out at the beginning/end, you'll know when these are happening as Stars appear at the top middle of the screen.
There are also various in-battle QTE which while not counted, generally stop you from taking massive damage/dealing out extra.
This Mission has a single battle against Orochimaru. Probably the easiest of the Early battles to S rank along with Battlefield part 1. Fling Jutsu at Orichimaru often and swap back and forth with the Characters for boosted combo damage.