No the Express will run sans battery BUT don't do it because starting it without a battery will blow pretty much every bulb on the bike. The battery is the voltage regulator on the early Expressi.
As for why you have no spark thats pretty uncommon on an Express. Truly those things run forever.
First things you are going to want to do is swap the spark plug (B6HS if memory serves me but look it up because my memory doesn't serve me often) and go in and clean the breaker points.
If that doesn't do it then you are going to have to chase the spark from the stator all over. You could just hard wire it to (stator to coil to plug) to see if its a wiring issue.
There is actually a guide written specifically to resurrecting a neglected Express.
http://mopedriders.org/article_view.php?faq=2&fldAuto=8
Expressi are kickass bikes and some cheap and easy modifications can get them up near 50mph.
I built an upper 40s (47mph top speed on the flat GPS confirmed) for a little over $300 including the cost of the bike.