Hell, I didn’t think it would be this easy!
Tag sneak, lockpick and speech. Use Good Natured and Gifted traits. Set strength and endurance to 2, perception to 7, everything else to 9. When you reach level 3 don’t take a perk until you have a couple of Buffouts. Then get the Strong Back perk. Take Silent Running at level 6. Forget about other perks, you don’t need them all. Take Pickpocket if you wish or Bonus Move to make running away in combat mode easier. Develop Science and Repair in addition to the tagged skills (books help, and your Good Natured trait makes them cheaper). Don’t wear armour, it’s too heavy. Even the power armour gives you +75 carry weight while weighing 85 pounds, unlike Fallout 2. Don’t get Dogmeat, ditch all other followers after recruiting them.
Sneak past rats in Vault 13. Take everything in Shady Sands. Go to Vault 15 and sneak through it (first difficult point, especially without silent running).
Get back, go to radscorpion caves, steal a tail (!) from any radscorpion, then blow up the cave entrance. If you think that’s silly, imagine your character cutting the tail with a knife then running away quickly. Radscorpions are so slow after all.
Go to Junktown. Take Gizmo’s quest to kill Killian, then go to Killian to get the quest to collect evidence on Gizmo. Go it and refuse to fight. Killian leaves Gizmo alone for some reason, but that’s his problem.
In the Hub forget about .223 pistol quest. Not that you need it anyway. However, save the brotherhood initiate. Just run away to the nearby cops for help. Hide behind them and as soon as thugs accidentally hit one of them, run away. The cops will do their job of keeping the peace and everybody’s happy, including the prisoner. To do the missing caravans quest, sneak along the northern cave wall to speak to the mutant without the Deathclaw noticing. This is the second hard point.
Go to Necropolis. Sneak past ghouls and rats (the third hard point). After you repair the pump and get the water chip, optionally talk to Harry to get to the Military Base and blow it up using the scientific approach. The just run away. But don’t forget to pick up the robes from the hospital in the Hub before going to Necropolis.
Now take the chip to the Vault. Go to the Brotherhood. Do the Glow quest, but don’t forget to disable the security bots right after turning on primary power. That means you should turn the power using Zax, not the terminal near the generator.
If you haven’t blow up the Military Base, then do the Brotherhood scouting quest. Then go blow it up. Sneak-talk inside, sneak-talk outside.
In Boneyard you’ll have to sneak past the Deathclaws to pick up the parts. It is the last hard point. Sadly, the Blades-Regulators quest is impossible to do without killing the Deathclaws. So not much to do there.
In the Cathedral it is the same sneak-talk. Don’t use Morpheus to get to the Master because it would be rather hard getting out without the Psychic Nullifier, unless you have the Mental Block perk which you don’t.
That’s it! Everything else is done the usual way. Just a few hard points and missed quests and the game is done. I wish they made Fallout 2 and Arcanum this way. In Fallout 2 you’ll have to skip a lot of quests, in Arcanum it’s almost impossible (think Black Mountain Mines).