Sakura plays this matchup on the ground. Good Giefs won’t jump at you because they have no reason to, but if they’re jumping obviously anti-air that shit. Gief’s pokes beat yours overall (st.mp and cr.lk most notably) so you need to be careful about what normals you throw out there. FlashMetroid just recently switched his main over to Zangief so I’ve been getting a little bit of practice in this matchup.
I still don’t have all the particulars down yet, but so far what’s really important is keeping Gief away from you (as is everyone’s matchup vs him). In order to do this, you have to play a very strong footsie game against him. Buffering cr.mk into shou is sometimes ok, but be wary of what range you do it at as he can whiff punish with cr.lk xx EX Green hand. Additionally, from certain ranges he can LP SPD a whiffed cr.mk thanks to his new range in AE. St.RH is a decent poke as well, but gets beat clean by st.mp so don’t throw it out too much.
The new far st.mp being cancellable allows you to buffer HP shou into this move as well, which can prove to be slightly useful as well. Use st.lp and cr.lk to counterpoke his own cr.lk and cr.lp buffers into EX green hand. If you are confident in your buffers as well, you can buffer both of these moves into HP shou and if you get the hit you get ok damage.
Gief’s mixup game changed drastically in AE, and with EX green hand no longer knocking down the matchup, in some respects, got easier. The hardest part about this matchup in Super was that if Gief whiff punished anything you did, which is considerably easy, he did it into EX green hand which gets him the knockdown. From there it’s just the stupid guessing game.
If you have your links down, and you definitely should, you need to turn any cr.mk xx HP shou into big damage by FADCing into cl.HP. You need to convert those small moments into big damage to keep up with Zangief’s superior damage.
Don’t bother trying to reset Zangief, there’s no reason to. You don’t want to rush him down and he has several options to counter your resets, so if you get an EX tatsu go for the otoshi and get the hard knockdown. Charged fireball mixup is safe on him since I don’t believe his Super or Ultra can grab you from that distance, but I have been known to be incorrect so test this for yourself.
I hope this helped even though it’s cluttered and none of you read what I write anyways.