I've beaten Ultimate Vault Hunter mode twice now. All I can say is take a Double Penetrating Unkempt Harold with you. That gun scales remarkably well compared to everything else until about level 56.
Also, try not to think about your gear too much before you start. The majority of it doesn't keep up very well past 53.
Also, make sure you take advantage of Quest reward weapons. Unlike TVHM you don't need to worry about getting a max level one as you can restart the story at any point in the game.
Last but not least, be prepared to see a drastic change. The enemies will kick your butt if you don't play strategically. Unless you're Gaige because let's be honest. She's easy mode.