Gonna assume you're coming back to the game after a long break.
Upgrading from 99 to 119 requires 300 Plutons, and having met the requirements set by Oboro in Port Jeuno (near the Qufim exit). The requirements are any
one of the following:
- Clear Delve once: Entry is easy, but getting a group can be difficult. People shout for it regularly, but often looking for support jobs like BRD. It is possible to get through this by mooching a win, though you may have a hard time finding people gracious enough to let you tag along.
- Clear a High-Tier Mission Battlefield on Normal difficulty or higher: Revamped BCs that require merit points to gain access (and having cleared the original fights). Some of these fights are much easier than others, though much like Delve, it might be tricky to get on board for one as a mooch, more so because of the merit point cost for entry.
- Clear a Sacred Kindred Crest BC on Difficult or Very Difficult: Revamped BCNMs in places like Horlais Peak, etc. Probably the easiest thing to get people together for, since entry is easy, takes very little time, and nets plenty of "Rem's Tales" chapters for all, which people use to upgrade AF/relic armor. I recommend shouting for this as your best option, there are always people out there needing more chapters. This is also the easiest method in terms of the difficulty of fights themselves.
Whatever you end up doing, which I'm guessing will involve shouting, be patient, and above all courteous and you'll be much more successful in making progress.