Winning an Ultimate is about staying alive and doing enough damage to finish the event with all optionals (in most ultimates the optionals are actually required).
Here are a few tips to help you achieve this. Many items are linked to the wiki so if you are unfamiliar with them please follow the links.
Run the Temporal primary specialisation, this will give you CONTINUITY which is effectively a second chance every 3 minutes. I suggest Strategist as a secondary specialisation to help with threat control.
However, for Korfez, it is important that some of the team run the Constable specialisation in order to negate the VSS Implacable's massive regen capability.
Bridge Officer Essentials
A very good contingency to use is Reverse Shield polarity. This will give you a real chance to escape or finish off an opponent. You should team this ability with the Fabrication Engineer Duty Officer to extend its duration by 8 seconds.
Getting out of trouble fast requires Emergency Power to Engines and Evasive manoeuvres. Make sure you always have EMPTE slotted in an ensign engineer station and combine it with the Emergency Conn Officer which will reset the cooldown on your evasive manoeuvres when you use EMPTE so it's ready most of the time. Consider slotting Deuterium Surplus in the device slots for that extra kick.
Not essential but very useful for neutralising DOTs (Damage over time) such as the Borg plasma dot, which can finish you off even if you survived and escaped the main attack. An ensign level is good enough for this, and it also activates Spore Infused Anomalies if you're an EPG build.
You really need to be seeing resists of 40% across the board for most ships unless you have very high speed such as in pilot ships, the higher speed increases defence stats. Boost your hull by using Jevonite Hardpoint devices. Boost your other stats by slotting the Tac/Eng/Sci buffs from the fleet starbase.
You should be using as many unconventional systems proc abilities as possible if you are using universal consoles.
This is going to be annoying for some of you but one of these consoles is a must-slot if you have it and that console is:
Not only does this give a massive boost to health very quickly but if above 80% hull you will gain from a staggering 40% BONUS damage capability and with unconventional systems this console can be ready again in about 40 seconds. Quite simply this is still the single best console in the game and you should do everything in your power to get it.
If you can't get the DPRM or already have it then the Protomatter Field Projector can give you and your team mates a massive heal, sadly this was also from an event which means it's impossible for some people to get. So folks, don't miss events, you never know when a BIS console is going to appear.
Easier to get consoles include the M6 computer and the DOMINO console, the latter requires an epic Phoenix token to get the Bajoran Interceptor from the store and is character bound so make sure you open it on the correct toon.
If you can't access these then get the Neutronium alloys from the Fleet Mine to help boost resistances.
Currently, the ship set of choice is a combination of the following for most builds:
Deflector: Fleet Colony for the extra Crit
Secondary Deflector: Also from the Colony
Engine: Competitive (get the right one depending if you're energy weapons or science)
Warp Core: Usually thoron or plasma infused from the Fleet Spire unless you need a specific set bonus.
Shield: Tilly's Experimental Shield (Disco reputation)
The above is a general selection of very good equipment but some builds will require very different setups, so use that as a rough guide.
It doesn't matter too much what build type you take into an Ultimate challenge as long as it has good survivability and good DPS. However here are a few general tips:
Notwithstanding the advice above, if you are using EPG then never drop gravity wells on Tzenkethi ships, all you do is pull them together allowing them to reinforce each other's shields and effectively make them invulnerable. For Tzenkethi use tractor beam repulsors to separate them and allow them to be picked off by the beam overload and cannon builds.
Tank: We are not interested in nanny tanks here but tanks in the purest sense of the word, these builds will attract threat and soak up damage whilst issuing good team heals, so they are more of a tank/healer combo.
Cannons: Scatter Volley is one of the highest DPS builds in the game at the moment and is great for clearing out mobs and targetable torpedoes. Rapid fire is amazingly powerful when built right for chewing through high-hull targets.
Beam Overload: One of the most powerful builds nowadays and like cannon rapid fire excellent for taking out the juggernauts, gates, tac cubes etc.
Fire At Will
Not recommended as you will pull the threat off the tank and most likely die, but if you can survive and it's built well can do similar damage to CSV.
The meta for absolute carnage in the battlespace, just don't get caught in your own splash damage.
Drain: Don't bother
Carriers: Fantastic if you have the new carrier advanced consoles (at least 4), Superior Area Denial and Scramble Fighter traits with Command authority and the Communications network console for fast ranking, you can make an off-tank with a carrier that can deal damage from a distance and maintain structural integrity. My favourite for this role is the universe class but many other ships can do well, obviously only twin hangar full carriers.
There is much more to be said, but this is a brief guide which hopefully helps you with your thought processes.