Depends on how you drive the car. 350 is gonna kill your clutch pretty quick but the tranny should be fine. As Dakoda said, you'll probably need a new differential to put that power to the ground. Also, what kind of torque is coming with all this hp, torque is a major factor on trannys, not just horsepower.
No question about the differntial, the car will NOT be driveable on the stock one. Play you would tear through the rivets in a second. 6puck 228mm clutch and l/w flywheel w/ a good LSD and you will be fine forawhile. Syncros like to go on the stock power so you may want to beef it all up.
Horsepower doesn't kill trannies, but torque sure will
. Like the man said getting that stock diff out of there should definately be a priority. Now where trannies get broken is when you have a non-loaded state and a instant load is presented....such as dropping the clutch to burnout or launch or burning out and hitting a height change in pavement. Really your trannies life is dependant upon how well you treat it. Making sure you have a gear before letting out the clutch and easing a launch out of the hole will get you alot farther than dropping the clutch all the time. Once you hit 300+ club nothing is a given anymore including your trannies lifetime