id start at a pawn shop since you dont have a real big budget. luckily you did say the power wire is ran, so that will save you money.. id buy an amp kit from rockford which are around $40, then a pawn shop amp with around 500rms, try for name brand. be careful at pawn shops, they try to screw you! around $80 for this, then find a sub box that comes with subs likeAmazon.com: Kicker Comp 07DC104 4-Ohm 2 Subwoofers In Vent Box: Electronics
all of these should keep you inside your $350 budget