General assumptions: Accessing the arena through the door unlocked with a keycard equals with the need to eliminate Sanchez in more traditional way. The game, of course, offers several different options on the implementation of this plan.
The most obvious method to eliminate Sanchez is to shoot him with a pistol or other standard weapon. I do not recommend this solution, because despite completing your objective, you'll get the attention of nearby characters and many of them have firearms.
Slightly better idea is using a sniper rifle and in order to do this, you have to get into the armory described in previous subchapter (screen above). As for reminder, the armory is located on the upper level of the arena and you can get inside through two locked gates (you have to use a keycard to unlock them).
Ilyon R700 sniper rifle can be found in the right room, with a guard sitting on the couch. So you should use Instinct or eliminate nearby factory guards.
Once you get the sniper rifle, you have to find a suitable point to shoot. The best choice is the side room, shown on the above screen. You can get inside through the door, which must be unlocked with a keycard.
Once you get there, approach the shutter and shoot the Sanchez in the right moment. Note - if you want to decrease the risk of detection after the shot, first eliminate all guards on the upper floor of the arena. It is also worth to hide inside the closet after successful action, to wait until the situation calms down.
Another method of eliminating Sanchez is using explosives. If you have remote explosives obtained in the previous stage, you can skip to the next step. Otherwise go to the left room in the armory (screen above). You'll find there remote explosives and C4 bricks. The main difference between them is that C4 bricks explode automatically few seconds after planting them. I personally recommend using remote explosives, so you can precisely set the moment of explosion.
Once you have explosives, go to the upper balconies. I recommend placing yourself at the place shown on the above screen - near the railing next to the VIP box entrance.
Throw the remote explosive or C4 bricks inside the cage, where Sanchez fights with Patriot. Depending on the explosive material, you have to use the detonator or wait for the explosion. If you use remote explosives, make sure that Sanchez is close enough to the charge.
The least aggressive method of eliminating Sanchez is arranging an accident with the arena lights. In order to do this, you have to get to balconies located at the top of the arena. This is the area, which only factory guards can access. One of passages leading to the arena rooftop is shown on the above screen. On higher difficulty levels it is observed by one of guards, so it would be best to distract him with a throwable object. You also have to have a keycard.
Use the stairs, but stop on the landing. Depending on the difficulty settings there are two or three guards in the area. I recommend distracting them and then using Instinct (you need to wear factory guard disguise).
Another passage leading to the arena rooftop is located in the VIP box. The entrance to the room is shown on the above screen.
It is best to get inside the VIP box wearing factory guard disguise. Additionally you have to use Instinct, so you won't get exposed by a single guard standing on the left. Eventually you have to reach the door shown on the above screen and unlock it with a keycard.
After reaching the most upper balconies, watch out, because depending on the difficulty setting these are patrolled by one or two guards. You can wait until guards walk away or eliminate them one by one, hiding their bodies in the crate or in the closet.
Your current target is a mechanism shown on the above screen, which lowers the lighting boom. Pull the lever, thus killing Sanchez (and Patriot).
After a successful action you have to leave this area as fast as possible, because local guards are already alerted by your actions. It would be best to use the passage leafing to the VIP box mentioned earlier. After getting there look around for ready-to-wear Patriot's entourage disguise (screen above) and use Instinct to safely leave this place.
Note - you can now scroll down to the The Arena - Exiting the arena subchapter, describing how you can end current mission.