Is it possible to solder the back pins of an Arduino board?
Yes, it's a standard through-hole header. You'll want to melt the existing solder to insert your wire. As long as you don't short two pins together it won't cause any problems.

That being said, I can't think of a situation where this should be necessary. Given that you're thinking about this in the first place, I'm going to hazard a guess that you're using a spool of wire and sticking the stripped end into the header. What you really want is some jumper wires with a connector that will fit snugly into the existing header.