Is it possible to modify a 97 impreza L drivetrain(hardened gears, cutch,etc) so that it could withstand the abuse of 350hp for reasonable time, 30k-50k? If it is possible, is it cost effective compared to swapping in a whole wrx/sti drivetrain?
I'm hoping fortifying is possible, only because ripping out the whole drivetrain scares me.

IMHO, Y, N, and you (or the shop you paid) would have to do more ripping out fortifying your existing 97L than to swap in STI. Good luck.

Yes you can. You can throw a RA gearset in your case with a decent clutch and you should be fine. There is no point to put a WRX drivetrain into your car, as your tranny right now will hold as much power as a stock WRX. A STI 6-Speed is a great tranny, but it is way too expensive.
Please let me know if you have any questions.
Thanks
Rick