The life time warranty solution might be easiest.
However, if you're still interested in how you might go about tightening the spool, what you can do is pull the belt from the spool a bit, remove the spool from it's locking mount, and then hand roll the belt back up. After that, put it back in the mount and you should have a tighter spring. Repeat as necessary.
If the shade won't stay up, the spring is too loose. Pull the shade down enough to turn the roller a few times; if it's extremely loose, pull it down about halfway. Lift the flat-pin end of the roller out of its bracket. Then roll the shade up by hand, keeping it tightly rolled. Set the roller back on the bracket and try the shade again. If it still doesn't stay up, repeat the procedure.
HowStuffWorks "How to Repair Windows: Tips and Guidelines"