Warrior can solo stuff, but it is a long process. I play a linked f2p, so I had access to most of the random dungeons, but if you truly wanted to do f2p warrior by yourself from the ground up, I think it would be possible but definitely a long grind. You have to build up your gear slowly, getting low hanging fruit first and then building out from there. Start with stuff like the pandaria vendor greens and TBC quest rewards with sockets, BfA rare procs (and socket procs from the BfA dungeons if you're willing to use dungeon finder for those). Then piece by piece, find TBC socket gear that drops from easier/earlier bosses, and swap out your existing items. You will probably need to enchant and gem your placeholder gear, which makes it even more of a grind. I'm not sure you can solo absolutely everything available to 20s, but I do know you can solo the stratholme service entrance (just barely) once you have basically-BiS gear. From what I've read, ramstein is generally considered the hardest 20's dungeon boss to solo. So the ceiling is there, its just a question of if you can get there without help.
Prot is going to be the spec of choice generally. Ignore pain is going to need to be up near 100% of the time. Healing sources are essential because you are a warrior and can't heal yourself on demand. For trash clearing/bosses with adds, victory rush will help carry you. It goes without saying that Crusader is necessary. Spinal healing injector is very important.
https://www.wowhead.com/item=137538/orb-of-torment?bonus=1826:1472&ilvl=25 is going to be necessary for harder bosses. Whether you can solo the boss that drops it before you get it is probably make or break for large scale warrior soloing. That said, you might be able to scrape together something reasonable even if you can't get the orb.