Apply both effects simultaneously
Feeblemind says (emphasis mine):
The target takes 4d6 psychic damage and must make an Intelligence saving throw.
The "and" tells you that both effects happen to the target. You can apply them in either order, because barring some other effect, one will not affect the resolution of the other. If you were meant to apply one effect after the other, the spell description would include 'and then' to indicate which effect came later (see, for example, the importance of 'then' in the answers to Does Chain Lightning deal damage to its original target first?).
Consider the spell description of lightning lure (emphasis mine):
The target must succeed on a Strength saving throw or be pulled up to 10 feet in a straight line toward you and then take 1d8 lightning damage if it is within 5 feet of you.
Here it is clear that a failed Str save first pulls the target toward you and then, after the pull is resolved, does damage (if the target is close enough). Had the spell simply said, like feeblemind, that the target is pulled 'and' takes damage then both effects would occur at the same time and could be resolved in either order.
As you point out, the damaging effect could make the target unconscious. However, unconscious creatures receive unmodified Intelligence saves, so the order of resolution would still not matter1.
From Conditions (Unconscious):
The creature automatically fails Strength and Dexterity saving throws.
As another example of a spell that does apply first one effect, then another, we have disintegrate. While it does not use the word 'then', it clearly indicates that the spell has an additional effect only if the results of the first, damaging effect leave the target with 0hp:
A creature targeted by this spell must make a Dexterity saving throw. On a failed save, the target takes 10d6 + 40 force damage. The target is disintegrated if this damage leaves it with 0 hit points.
Since Feeblemind neither includes 'then', nor specifies that anything special happens at 0hp, we can conclude that the save for the feebleminding effect is made independent of, and regardless of the results of, the damage.
1The order of application might matter if the damage killed, rather than rendered unconscious, the target. This might happen if the target was an NPC that died at 0hp, if the spell was cast on a creature already making death saves so that the damage caused a fatal fail, or for a creature whose maximum hp were so low that the damage caused instant death. In this case death should not protect the creature from what is supposed to be a simultaneous effect, so the save should be required before the damage is done. Further, note that spells which restore a creature to life specify which conditions and effects they remove, so it is reasonable to assume that the target would still be under the effects of the feeblemind even after its death and later revivification, raising, or resurrection.