I would actually recommend you to take the train that leaves Basel SBB at platform 8 on 10 June 2007 (Sunday).
Basel SBB (00:08)
Zuerich HB (01:17)
You may continue to the airport with a taxi from Zuerich HB (book the taxi well in advance and confirm on Saturday).
After safely reaching airport, you can relax a bit and check-in.
If I was you, I would certainly do this rather than spending 350 bucks
Nevertheless, I wish you a safe journey !!