What even is a “Rush Mission” to get a title?
I tried doing the monument’s RQ (Repeatable Quest) but still no title after 5 times.
How exactly do you do a rush mission?

I quess it is the same as Defeat X monster. There isn't certain indicator how many we've to defeat to get certain title, but I achieve those easily leaving game on sleep mode while phone charges next to my bed and I'll sleep 4-8h at night.