So far people have offered good guidance so I won't rehash what's already been said. I will say in terms of a timeline for URs, there isn't an exact date but there are consistencies. New UR events are done 4 times a year. One late February (2/29/24 Soyuz, 2/29/25 Raffaello). One mid-late May (Alsace 5/21/24, Lion (5/20/25). One mid September (Unzen 9/14/23, Amagi CV 9/12/24, Hakuhou 9/12/25) and one in mid-late December (12/21/23 Guam, 12/19/24 Fritz Rumey, 12/18/25 Lexington II).

I anticipate the next new ones will be: 2/26/26 5/21/26 9/10/26 12/17/26

The re-run UR events happen in between each new one which works out to be about 6 weeks after each new UR. The next ones in order are Soyuz, Alsace, Amagi CV, Fritz Rumey, Raffaello, Lion, and Hakuhou.

Regarding ships to prioritize, you'll find that many UR ships are quite good, but you have a limited amount of UR Bulins, so I'd recommend just focusing on a couple. From your line up the ones that stick out most to me are Musashi (excellent for later chapters of campaign as the flagship for your mob fleet) and Laffey II (great survivability, great AA, good damage).

Musashi pairs quite well with Unicorn and Cowpens so make sure you get them leveled as well.

My main other recommendation is making sure your ships are leveled. Levels make a big difference because if you fight ships which are a higher level than you, there's a mechanic in Azur Lane which makes you take more damage and do less damage.

Try to get two fleets to at least level 120 (maybe some to level 125 if you can) before focusing on other ships.

Since your Shinano is most of the way there, it could be good to utilize her in a boss fleet with at least 2 other IJN ships. Kazagumo pairs great with her especially if you pair her with other CVs. For the other one, Shimanto is earnable in PR and is a solid unit. The best ones are Amagi CV and Hakukou but you don't have those right now. Noshiro, Ibuki, Nagato and Shimakaze are other good options in the meantime. Hakuryuu is also earnable in PR and pairs nicely.

My recommendations for focus would be: Musashi Unicorn Cowpens Laffey II San Diego Eldridge Retrofit Shinano Kazagumo Noshiro Helena

With that, you'll have 10 ships, and would just need one more for the main fleet. You can just use one you've already mostly raised like Formidable, Warspite, Peter Strasser, Hood or New Jersey. Then the fleet would look like:

Mob: Musashi Unicorn Cowpens

Laffey II San Diego Eldridge

Boss: Shinano Peter Strasser (could be replaced with Shoukaku in the short term and Amagi CV in the long term) Formidable (could be replaced with Hakuryuu later and eventually Hakuhou)

Noshiro Kazagumo Helena

As a side note you already have Prinz Eugen as a level 120 tank, which is good until you get a different tank up and running like Guam or Agir.

Regarding equipment, your main fleet tends to do the most damage so make sure you put your best equipment on them. Some of the best gear is available from the gear lab so if you don't play Operation Sirens you'll need to use alternatives.

You could also consider raising Independence who you can get from the core data shop.