Some more details might be useful, I feel. We can know the details of the current mission pass here.
For example, if you are going to play male CAST, you'll get Drake set as gold rewards of tier 10 and Regia set as gold rewards of tier 29. If you need one of each set, then these will be all. But if you want 2 head parts ticket to use them on multiple characters, then you'll probably need to complete the extra tiers to 35.
Then, going to whether you can complete this in the given time window. I suppose it will be difficult without purchasing tiers with SG. If we assume the common guess of May 26th being the PC release, that means there are only 2 weeks left within this season. If given 3 weeks, I think anybody will be able to finish most of the full 35 tiers, but 2 weeks are too short. You'd better expect to buy several tiers for the completion (buying a tier costs 25 SG each).
Also, all CAST parts are gold rewards, so you'll need a Mission Pass Gold Ticket too, which costs 200 SG. By completing all tiers, you can get a total of 200 SG as mission rewards, so maybe we can take this as a refund (and to be used to buy the gold ticket of next season...).
So I think you'll need to put a decent amount of SG earned from the story related ARKS missions, unfortunately.