First of all, I'd like to note that you should be saving as often as possible and always have at least two save files in case you accidentally save with an enemy behind you or whatnot. Ultra-Violence can be a bit of a challenge, especially in the later episodes, but you can make this much easier for yourself by mastering strafing, sprinting, and shooting at the same time, as well as keeping a good distance from your enemies (unlike in Doom 3, you can easily kill with the shotgun from very far away). There's rarely a reason to stop sprinting during a fight, and it's also a big part of what makes the action in Doom so fun.
Here is the most useful guide for getting through the maps quickly (just use the table on the right side of the page to select the next map): http://doom.wikia.com/wiki/E1M1:_Hangar_(Doom
Just follow the "Essentials" sections of that guide and you'll take the quickest route to the exit. However, I recommend going out of your way a bit to get ammo, health, and armour as needed (see the Secrets sections of the maps if you're having a hard time finding weapons/ammo). If you've already completed Doom 3 on Nightmare, you should have no problems at all with this, especially if you've played the originals back in the day. Good luck!
Note: You will likely have a difficult time with Episode 4 maps 1 and 2 (hardest levels of all 3 Dooms honestly, discounting Plutonia Experiment) due to the insane number of enemies and there being almost no health/armour or ammo (especially in 1). I highly recommend referring to this playlist for help with these levels and any other that may prove to be problematic for you: