I started testing this game in june. My first impress was 'WOW. It's amazing!' and then I imagined that why don't I do a speedrun?
So I was trying, playing looking for glitches day by day. The game itself is not hard. You don't need to watch your health bar or your ammo if you play on easy.
You cannot go fast. Nothing boost you just monsters. You are not able to jump over things to reach the map changer trigger as fast as you can.
The game has weird scripting and things aren't working as you remember from your last try.
I would be glad to render thanks but... nobody helps me. There are only two person but at two point of the map that means every glitch, trick that is well known from now was my idea.
The following segments are named by savegame names.

l00_intro:
Firstly, I run through Miller and quickly open the gate. You need to pick up three ammo and a medkit.
I was quick therefore Miller is teleported to the next event. I go up and kill the mutant. After that I have to be very quick.
I kill the first mutant and then I riddle the next two with bullets because if I am not enough fast, another mutant appears. (I'm not sure)
Then I run toward the tower to get the cutscene. (thanks to ZhouTai)
Finally I must wait. There is no chance to finish it faster.

l01_hunter:
I go up to the hall and kill the mutants. The first group goes front the ceiling, the next one is from the left air-duct and the last scripted mutant goes from the right.
After that they come from random places but if you are lucky, there won't be so many. I think it doesn't depend on how many mutants you kill.

l02_exhibition:
I just have to be quick to avoid sharp edges. You can go through the first man if you choose a good angle but the second man block you whatever you do. At the gunstore, I am stucked for half a second but the shopman starts speaking that means everything goes right.

l03_chase:
There is no place for improvement. The only thing that can happen is a zoom effect (after you come out the cover) that moves you 2-3 meters toward.

l04_riga:
I pick up the money, run to Bourbon and after discussing I go next to the toilets because Bourbon would stop if you were not there.
Then I buy 3 grenades. I try the red ammunition box but there was no ammo in my video. But there are rarely existing.
The level ends when Bourbon goes through the door so I didn't have time loss.

l05_lost_tunnel:
I had four grenades. It is enough for bandits. I go through the tunnel and wait for Bourbon. I missed the wire so I had to redo it but I didn't miss any seconds.
One grenade for the camp, one for the guard and the last two or three are easy to kill. I lost a second because the last door opens automatically and I didn't recognise that.

l06_bridge:
This is the only place where I had bug. My gun stayed on the screen and the other one also appeared but It didn't make the run longer so... I kill every mutant and bypass the train for... fun. After that We know We cannot jump to the door. I presented It in the Metro 2033 thread.

l07_catacombes:
I collect all the ammunition. After the dreamy scenes, I reach the big door. Bourbon arrive and We start shooting the mutants.
I don't know how is it timed but I have no idea to shave off time. You have to be with him but shooting skill doesn't help.
There you have to crouch and go through the big door but this information is come up after I uploaded the video. (thanks to kekskiller)

l08_market:
I run to the shop and imitate that I buy something. It moves Bourbon to the next event. After that I go to the gate and wait for Bourbon.

l09_dead_city_1:
I don't waste my time jumping on the car so I go to the end. This is a big place but I cannot find any improvements.

l10_dead_city_2:
The screen is black but I memorized the way to get outside. (skillful trick uh..?) I walk to the truck and wait for the demon.
I throw a grenade and hopefully It kills it. This trick skip a little section of the following events.
After that I kill some mutants and climb up to the fence. You can see I do this because I can reach the well quicker.

l11_dry:
Easiest segment. I just found the best way to get out of there. I don't need to kill anyone or avoid shots.

l12_ghosts:
The first two ghosts can be avoided but the guard cannot. Wait for them to move to the sides. After that I just run.

l13_anomaly:
At the end, the anomaly hit the door and I run to the railcar. It force khan to jump to the next event.

l14_cursed:
This is a very ugly part of the run. I did everything correctly but It seems there is another improvement but I didn't know anything about it.(Kekskiller -10sec)

l15_armory:
Nothing special but I bought new kevlar clothes for the next level.

l16_frontline:
Very primitive way to run through the map but I cannot find any quicker way.

l17_trolley_combat:
I cannot bypass Pavel so I missed 1-2 seconds.

l18_depot:
I must kill the Nazi to go through the station silently.

l19_defence:
Very annoying part. I tried everything but the only place where You can clim up is the top of the iron box where the boss stays. So I must fight.

l20_child:
It's hard to move the mouse but there was no problem.

l21_nazi_outpost:
I choose the quickest route and It works well. Tip: Going out the map is longer with 10 seconds

l22_black:
'Sneaking through the shadow...' Quickest route, no problems.

l23_polis:
I have to stay to Miller because It starts the next event.

l24_alley:
Nothing special.

l25_library:
When I am in, I have to run to the door but there is a Librarian, and It block your way. You cannot cut the route.
After that I shoot the luster to skip the demon cutscene.

l26_archives:
Nothing special.

l28_depository:
First and last ladder jump works well. Then I go through the map as fast as I can.

l27_driving:
Nothing special.

l30_darkstar:
Nothing special.

l31_catacombes:
Nothing special.

l32_cave:
Nothing special.

l33_d6:
Most boring and longest part of the game. You cannot run down to the lower floors. The generators broke down randomly.
Your mate stucks everywhere at the lowest floor. So I mustn't kill the mothers of the giant ameoba because He would be just standing.

l34_biomass:
The floor is covered with annoying oranges and It is the true chaos.
After that I use the crane and it looks bad but I lost just one or two seconds. And I was hesitating after It, but I was just careful.
You can quickly fall down and die.

l35_division: Nothing special.

l36_ostankino: Nothing special. The same as the first segment but then You have to wait for Miller sometimes.

l38_tower: Hardest part of the game. You can stuck in everything and It's hard to navigate your mouse.

l39_ethereal: Nothing special. Happy End.