After you get Gwen into quarantine you will be in the city for the first time. After a few seconds of roaming, you will be alerted to a "disturbance" in the city, followed by a location to get to on your mini-map. Go there and you will find the S-01 messing up a park.
When the battle starts you are on the ground - immediately jump and start web-slinging around the huge machine beast. There will be points to web-zip towards (R1). After ever assault make certain you keep web-slinging (R2). If you hit the ground press start, and from the options choose "Restart" - you will be back at the beginning of the battle.
If you die and continue you will find yourself on the ground - don't worry. Just jump again and stay in the air. As long as you don't touch the ground after you initially jump you will be fine.
After you take out the 4 points on the huge Spider-slayer's body, you will then need to take out its eyes (QTE's). After that you need to web up the legs ("O" button). Just remember to keep web-slinging and you will be fine.
If I missed anything let me know in the comments below.