The final limit breaks for select SSR weapons is from a recent update. The list is basically the weapons traded in the shop for draconic eyes (aka what used to be the gacha non-kamihime weapons), the gacha kamihime weapons, union event SSR weapons, and the original advent event SSR weapons (not the second SSR weapon added in the 2nd reprint of an advent).

It does not cost draconic eyes. It needs Orichalcum. You get Orichalcum from completing X number of objectives during Guild Order type events.

What are Guild Order events? The latest in highest-difficulty-yet content; they run concurrently with the regular event schedule as you can only attempt them once per day, kind of like Ragnarok advents. They're kind of like advents in terms of being composed of multiple stages/battles, but they're 5 stages long as opposed to 3. Stages 1-2 are grunts, 3 is a demogorgon, 4 is grunts, and 5 is the boss.
Each Guild Order lasts 2 weeks long. The first week has a set of 10 objectives. The second week has its own set of 10 objectives.

What kind of things are these objectives look for? Let's look at Crom Cruach, the first guild order which also just got reprinted.
Week 1:
Challenge the guild order 3 times
Challenge the guild order 5 times
Beat the first stage 3 times
Beat the second stage 3 times
Beat the third stage 3 times
Clear the guild order with Arthur
Clear the guild order with at least 3 SR kamihime
Clear the guild order with at least 3 Offense type kamihime
Clear the guild order with the soul alive
Clear the guild order with at least 5 survivors

Week 2:
Challenge the guild order 5 times
Beat the second stage 3 times
Beat the third stage 3 times
Beat the fourth stage 3 times
Clear the guild order with Joan of Arc
Clear the guild order with at least 3 R kamihime
Clear the guild order with at least 3 Offense type kamihime
Clear the guild order with at least 1 of each type of kamihime (Offense/Defense/Balance/Tricky/Healer)
Clear the guild order with 7 survivors
Beat the fifth stage within 15 turns

So 20 objectives/missions to clear within two weeks. You get 1 Orichalcum after getting 6, 10, and 18 mission points. 2 Orichalcum for clearing all 20 for a total of 5 possible Orichalcum per Guild Order.

How much Orichalcum do you actually need to perform a final limit break? That varies on the type of weapon.

Draconic eye exchange weapons need 5.
Kamihime weapons need 3.
Pride weapons from Union events also need 3.
The original set of advent SSR weapons need 2.

Other materials are also needed, but they're easier to get in comparison. The next toughest material to collect is probably the 10-20 regalia.