If the N/S switch is just out of adjustment, (and not bad), you should be able to hold the key in the start position and move the shift lever back and forth from reverse to drive.
Across the neutral position, and the starter should work.
The alarm thing may be your problem. Can you put the key in the 'on' position, and then, using a starter jump botton (buy one),jump the starter and start the car.
Read:
http://www.toyota-supra.info/forums/...ead.php?t=9087