will it be enough

I have a complety stock C4 (72 model) case fill. I just bought a TCI master racing overhaul kit from Jegs TCI 529500 - TCI Master Racing Overhaul Kit - JEGS

Will my C4 handle a 350HP 351W with just this kit and a tranny cooler?

350 hp yes, IF the kit contains the parts to put in five clutches where there are four and IF the cooler is a BIG one.
Coolers are rated in lbs of towing weight so get one rated at 33000 lbs at a minimum. And IF your engine is naturally aspirated.