AR Training (1 of 4)
You will be informed of the fact that you can approach this side mission by a W marker on the city map #1. Head to the point marked as AR Training (1 of 4) in the south-east part of the city, nearby the Industrial District entrance. Look around for a red Batman sign on one of the towers #2 and approach it to activate the challenge.
The overall idea is rather obvious, as you have to fly though five green checkpoints #1 by using the glide ability (hold down A). Throughout the flight you will only have to change directions slightly and shouldn't have problems with reaching the last, fifth checkpoint #2. Be sure not to hit anything or miss any checkpoints, as it means failing the whole challenge (this is also true for the other challenges in this mission). In case you fail, you just have to return onto the tower to try again.
AR Training (2 of 4)
Now you can head to AR Training (2 of 4) #1, found on top of the large crane #2 in the Industrial District, so in the same area as the last challenge. Using the Grapnel Gun will prove very helpful in reaching the target.
Even though this challenge requires you to fly through only four checkpoints, it's harder than the previous one. Start off by flying through the first checkpoint #1 and at the right moment press the right trigger to activate the dive bomb and fly straight down #2. Use it to fly through the other gates and complete the challenge.
AR Training (3 of 4)
Find the AR Training (3 of 4) on the map #1. You'll have to reach the top of a crane in the Industrial District area in this case as well #2.
This time things will be even harder than before. Right after flying through the first checkpoint, start diving (right trigger) to reach two further gates #1. However don't rush it with diving, as right after flying through the third one you have to flatten out. Fly through the fourth checkpoint below the tower #2 and afterwards to the fifth and last one.
AR Training (4 of 4)
Head to AR Training (4 of 4) #1. It will once again take place in the Industrial District, though this time you have to reach the top of one of the towers #2. the Grapnel Gun will of course come in handy here.
Fly towards the first gate and afterwards start diving down. Flatten out right after flying through the second gate and reach the third one, making sure not to hit the obstacle on the way #1. Get ready to press the right trigger again, flying through the third and fourth gate. Flatten out once again and reach the rather easily reachable fifth checkpoint. It's on top of a small platform #2.
Note that by completing this challenge you have gained access to a very handy prototype Grapnel Boost. You won't however get it automatically, but will have to reach the drop zone added to your map #1, the top of the drawbridge in the Industrial District #2. I'd recommend checking it out at once (its description can be found in the Wayne Tech), as it makes moving through the city much easier.
Advanced AR Training 1
It's not the end of the AR fun, as there are four more challenges waiting for you. You can complete them in any order you like, but let's begin for the one marked as Advanced AR Training 1 #1. It can be found in the very middle of the Industrial District, on top of one of the towers with a neon sign #2.
Jump off properly and fly towards the first checkpoint. After flying through it wait a short while and hold down the right trigger to dive #1. Fly through fates #2 and #3 this way and afterwards quickly flatten out to fly into a small tunnel #2. It's important to start off as close to the ceiling as possible, as the last gate is pretty far away.
Advanced AR Training 2
Now you can search for the point marked as Advanced AR Training 2 #1. It can be found in the Bowery, in the south-west part of Arkham City. After getting there look for the Gotham Bank building #2. You need to reach one of the gargoyles.
The challenge is rather difficult, so I'd suggest staying focused. Glide towards the first gate and start diving right after passing through it #1. You have to flatten out the flight at the exactly right moment, thanks to which you end up below a roof and fly through the second gate #2, without touching the ground or any other objects.
Fly forward, but get ready to turn left at the first chance #1. Lean out the analog stick only for a short while, so that the turn won't be too sharp. Now you just need to safely reach the last gate #2.
Advanced AR Training 3
Open the map and choose the point marked as Advanced AR Training 3 #1. It's in a well known place, on top of one of the cranes in the Industrial District #2.
This time you have to fly only through three gates, but it doesn't mean it will be easy. Fly to the first one and start diving, turning a bit to the right at the same time #1. You will then have to flatten out fast and fly through the narrow pass #2. It's important not to be too low when you stop diving down, as otherwise you might now be able to reach the last gate.
Advanced AR Training 4
Find the last point, marked as Advanced AR Training 4 #1. It's in Park Row, in the district where you began exploring the city. Search for one of the highest buildings #2 and reach its ledge.
The challenge isn't too hard overall, but requires some skills. Fly to the first gate and start diving down to get through the second #1. Flatten out and fly forwards, turning around after a dozen or so meters #2. Make sure not to hit any nearby buildings while doing it.
You should end up opposite to the third gate #1. Fly through it, but make sure you're high enough while doing it #2.
The ground level here is rising, so reaching the fourth gate #1 would normally be impossible, as sooner or later Batman would hit concrete. As a result you have to use the Grapnel Gun and grab one of the objects BEHIND the checkpoint (not on the sides!) #2. Don't worry if it will force you to turn around once again, as the only thing that matters here is flying through the fourth gate. Doing so will end the mission.