Ulduar is one of the last raids Blizzard designed that really captured my imagination. It's a longish raid to solo, but I always try to run it each week for the chance at the Mimiron's Head helicopter mount from Yogg-Saron. I've condensed my run down to 5 minutes here so you can see the entire raid run in a reasonable amount of time.
You can also get lots of nice pieces for transmogrification (IMO, some of the coolest-looking in the game), and usually a handful of BoE epics and greens, pets and crafting recipes. If you take the time to kill all the trash after XT-Deconstructor (especially in Freya's room), you will probably make around 2000 gold or more on the stuff you vendor--plus whatever you can sell on the auction house. Several of the bosses will also drop the Fragments of Val'anyr needed to make the legendary mace Val'anyr, Hammer of Ancient Kings. There's also a meta-achievement you can do in Ulduar; complete all the achievements and you get another mount.
To go for a FAST MOUNT RUN (as of Legion): Make sure you're in 25-man mode. As soon as you enter the raid, use the teleport pad by the entrance to teleport to the Ulduar Tram (or Spark of Imagination if it's available). Kill Mimiron (no need to do Hard mode like I did), then use the teleporter near his room to teleport back to the Observation Ring. Clear the other Keepers: Hodir, Freya, and Thorim.
Once you kill the Keepers, do NOT talk to them to ask for their assistance on Yogg-Saron. You have to kill Yogg without any Keepers helping you in order for the mount to have a chance to drop. Go down the Hall of Memories, kill General Vezax, then continue down to Yogg's room and kill him. If you're lucky, the Mimiron's Head mount will drop, or a vanity trinket that summons a creepy tentacle. The mount only has a 1.2% chance to drop, and will only drop in 25-man mode with NO KEEPERS helping you.
If you're a little underleveled and the Keepers are taking you too long, you CAN kill them all on 10-man mode. After downing General Vezax, DO NOT KILL YOGG YET! Use the teleporter pad outside Yogg's room to port to the entrance of the raid. Go outside, set the difficulty to 25 man, and re-enter the raid. Use the teleporter again to go back to Yogg-Saron's room and kill him. Pretty much any DPS class over level 90-95 should be able to solo this raid with little to no problems.
At level 100+, most fights only take a few seconds. I'm using a lv. 105 retribution paladin in this video. There are some weird mechanics to be aware of in a couple of the fights though, so make sure you read up on Wowhead if you've never done Ulduar before.
Good luck!
Music: A Small Victory by Faith No More
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