How can I prevent this concrete block wall from falling over?
It's really hard to say without being there in person. The wall certainly looks unstable and ready to be taken down, but a lot of concrete walls can look terrible, but still be structurally sound. We owned a house with a concrete block wall reinforced with rebar that had perhaps a 10 degree lean outwards that looked like it'd fall any day, but it took a LOT of sledge hammering, chiseling, jacking, and gas-powered diamond-blade cutting to actually get it out. It likely would have stood for another 50 years on its own.

In your situation, it appears that you've lost all the earth around your footing. At this point, it may be a lost cause and you'd be best taking it out. Ideally, you'd then replace it with a stepped retaining wall using a dry-stacked block. The dry-stacked blocks are a) easier to install and b) when installed properly will allow water to drain through rather than build up behind the wall pushing it out. 

All that said, it appears that your neighbor has build a foundation right up along your wall. I don't know where you live, but I'd think in a lot of places that'd be against code (you typically can't build that close to the property line). In addition, it seems risky for them to depend on you to hold up their foundation with your wall. Seems that they should be bearing part of the ownership of whatever is retaining the earth on this property line. 

Finally, if that's a down-spout in the lower photo emptying between the two walls, then a) take it up with the city, as that is likely also breaking all sorts of code regulations and b) your neighbor really should be worried about that. You don't want a downspout draining near the footer of your new kitchen's foundation.