7.Genesis/a new hope
Check the bulletin board. You start with Matrix Level 1.
Exit the classroom into the 2F hallway for an event with Kotomine.
Talk to Sakura in the Nurse's Office to get your weekly item.
Go to the Arena entrance for an event.
Enter Arena 1F.
The side paths are sealed, so head down the only available route toward the end.
There is a Nephilim on the right side of the far west hallway as you head north. You can't miss it.
Right before the exit, you have a boss fight against a Berserker. He always uses a powerful attack Skill for his first move. Unfortunately, he has resistance to stun that causes Hack and Shock to only work part of the time, so use Guard or a defensive Skill to counter it. Other than that, he's pretty weak at this point in the game, so you should finish him quickly.
Talk to Rin(Roof)/Rani(3F).
Go to the Arena entrance for an event. Choose the 1st option.
Enter Arena 1F.
Head to the southwest corner of the map for an event. For Saber, choose option 2, then 1. For Archer, pick the 1st option. For Caster, select option 1, then 1.
You face SE.RE.PH, which is incredibly easy for a supposed boss. I would recommend spamming attack Skills for a quick victory.
Continue through the dungeon normally. Make sure you get the 1st Trigger to the west of the exit.
Go to your Private Room for an important conversation. For Archer, pick the 2nd option. The others have no choices. Afterward, you get Player Matrix Level E.
Talk to Rin(Roof)/Rani(3F).
Accept a quest from Taiga on 1F.
Go to the Arena entrance for an event with Leo.
Enter Arena 2F.
Head forward through the dungeon. Definitely make sure to grab the Teacher's Emblem from the eastern section because it can only be obtained on this day. Also, you should probably grab the Staff of Rebuke from the northwest, considering it's the strongest of all of the 2 Formal Wear with MP healing in the game.
There is also a Nephilim around the corner to the south of the Teacher's Emblem.
Advance through the south hall from the central room for an event.
Backtrack a bit to fight Leo.
Gawain is immune to stun from Hack or Shock, so scrap any ideas involving those. He always uses the same moves for his 1st and 6th actions. On the first turn, he always buffs on the 2nd and 5th actions. He may use an attack Skill for his 3rd move on either the 2nd or 3rd turn, possibly both. Now, this is the crucial part—make sure you apply some sort of buff before the end of the first turn. Doing so ensures Leo will never stun you with a Code Cast. If you see a Code attached to the 6th action on the 2nd turn, he's about to remove it, so make sure you buff again on that slot.
After the battle, you get Matrix Level 2.
You can't advance beyond this point, so backtrack and explore any areas of the 1st half of the dungeon that you haven't covered yet.
Talk to Rin(Roof)/Rani(3F).
Check the door to the Multimedia Room in the 3F east hallway.
Talk to Taiga on 1F for a quest update.
Enter Arena 1F. Not 2F, 1F.
Head to the location you fought Berserker in front of the exit on day 1 to pick up a feather.
Meet Rin/Rani at the Multimedia Room.
Go to the Arena entrance for an event with Leo.
Enter Arena 2F.
Explore the remaining half of the dungeon today.
Go to the end of the long hallway directly north of the exit for an event with Leo. Tell him he's wrong and fight him. You get Matrix Level 3.
After the fight, pick up the 2nd Trigger.
Make sure to grab the Tiger Thermos from the northwest corner of the southwest room.
Go to the Library for an event with Rin/Rani and Leo.
Give the Tiger Thermos to Taiga on 1F.
Enter Arena 2F and grind until you're satisfied.
Go to your Private Room and organize your information to get Matrix Level E.
Shock still doesn't work, so do not equip the Sword of Repentance. Maleficent Garments, on the other hand, is a great choice.
When you're finished preparing, talk to Kotomine on 1F.
Saber will always buff himself with Divine Aegis and Siege Perilous on the 2nd and 5th actions of the first turn. Divine Aegis is only ever used on the 2nd action, and Siege Perilous on the 5th. Both will always be used together. Siege Perilous makes his successful Guard counters stun you for the next move, so you should use Vanish Add on the 6th to keep things from getting out of hand.
He will sometimes use the attack Skill Blade of the Devoted for his 3rd action, so Guard if you're uncertain. Be extra careful if it's coupled with a Code Cast because Leo will sometimes lead it with a Code to make you unable to Guard against it. Of course, if you have Archer or Caster, you could always use a defense Skill. But with Saber, you are left with no real options if he does this.
Unless you never let him use it. As long as you have any kind of buff applied at the end of a turn, he will never use that Code. Instead, he will sometimes use Vanish Add to remove it. So if you ever see a Code Cast on the 6th action, that's what that is. Make sure you reapply a buff on that action so that he won't use his other Code and Skill combo on the next turn.
He will start using his Noble Phantasm when he falls below half health. Saber always says "Leo..." in a pained voice before he uses it, and he always uses it on the 6th turn. Guard or use a defensive Skill if available to defend against it.
When his HP falls below 10%, he will use Numeral of the Saint for his 4th action, which reduces the damage he takes by 90% for the rest of the turn. It will make things much easier if you kill him before he can do that. In other words, once you get him to 10%, quickly finish him in one go.
Unless he's using his Noble Phantasm, he will always use the same standard move for his 1st and 6th actions whenever he uses any Skill. It's also a pretty safe bet that he will do the same when not utilizing a Skill since it's rare for him not to.