No first hand experience with the 1jz. To get to 350 on the stock twins try a boost controller (manual or electric) with a bleeder or plain old shimming the gates. A front mount intercooler will allow for more boost. Stock fuel delivery will limit the amount of boost you may safely run. Maybe 14psi max? 350hp may be a reach without upgrading turbo. I'd recommend a single conversion. Check HKS for example. It will cost money. T04R would be nice. Of course, then you are way over 350hp and you will have lots of other upgraded items like exhaust manifold, feul system and management. The sky is the limit but with stock turbos, be ware they will break easy at higher boost.
Nice car, good luck.