Unfortunately, to my knowledge there is no such rule about which effects happen first. It's possible that the authors didn't foresee any situation where such a rule would be needed and they just explicitly tell how to solve some specific cases instead (the interaction between some energy protection spells in 3.5e comes to mind).
This means we need to look at how the effects have been described, i.e. we need to parse English.
This specific case is unfortunate: as written (or at least, as written in the SRD), the utility of the brooch is not clear at all. It absorbs magic missiles, but how? When? What's the in-game effect? It is not written. All we know is that a previous game (upon which PF is based) had the same item absorb magic missiles aimed at you.
I say this case in unfortunate because a better text could have given us some hints: does it absorb dame you took, damage you would have taken or missiles coming your way?
The shield, however, only reduces damage taken by you. If the brooch is absorbing the missiles before you take the damage (which is uncertain, but the most we can say by parsing the word "absorbs"), the shield property never activates as long as the brooch is working.