If I play Raucous Anthem, and then play Meticulous Restorationist from trash, will Meticulous Meticulous Restorationist be able to come in readied and with +3 might?

No — Restorationist enters exhausted with its printed 2 Might; it can't be readied or buffed by Raucous Anthem.

Raucous Anthem is an Energy 7 card, so Restorationist's trigger does fire, but the trigger waits until the Anthem has fully resolved (Rule 420.4.a). Anthem readies and buffs the units you already control at that moment — Restorationist is still in the trash, so it's not one of "your units" and gets nothing. Only afterward does the trigger go on the chain, letting you pay [C] to play it from trash, where it enters exhausted as normal (Rule 359.2.c).

The +3 Might is also a one-time "this turn" grant to the units present at resolution, so it doesn't follow units that enter later.