Once in the courtyard the way to go is to the left. Go through the archway and then turn left. Head straight and then through door on right. If you have time left on the Hammer of Dawn use it to kill the Troika gun emplacement.
After the Locust are killed and emergence hole will open. Kill the Locust that come out. You can use the Hammer to close the hole, if you still have time.
When those Locust are dead another emergence hole opens. Kill these Locust and close the hole. After the fight the door out is in the back right from where you entered.
Follow the hallway to the right and out the door here. Several Locust attack here. Kill them. When the Locust are dead a Boomer comes out a door.
Go through the door. Follow hallway to the right. There will be a door to the right Marcus can kick open. Open the door.
Exit the hallway through the door. Go straight ahead and to the right down a path next to building. Locust will start attacking from top of stairs. Kill the Locust.
After the fight climb the stairs to the top. Go either way around the building in front of you. On the other side will be a door.
Go through the doorway. Follow small hall to open area. In open area go to left. Follow this path. Keep going forward towards the arch.