Adapting charger for NXT 9798 battery
The two batteries and chargers are very different, in that the new one is 10V DC whereas the old one is AC (while it's supposed to be 9V, I think the transformer was actually 10V too).

As it's AC, the polarity doesn't matter, but I don't think using DC current is recommended (even if in theory the rectifier circuitry in the old battery should be able to accommodate it).

You should be able to buy a spare fairly easily though, whether at a local LEGO event or directly on BrickLink. Note that it's the exact same that came with 9V trains, so you may want to look for the complete transfomer + regulator as well, which may sometimes be cheaper to find.