Will this DIY window fan setup help cool two rooms in our tiny, narrow apartment?
If you have ANY traditional AC system this is a terrible idea, it is allowing a significant amount of conditioned air to naturally transition outside. 

If you already reliant on natural air flows and fans, can time it right to feed cool air in, and use some insulated panels to manually block either/both sides not in use it will help. Also, fans are typically (manufacturer specs can greatly differ) more efficient the larger they are, the longer the blades the more curved they get because they are laterally spinning faster at the same RPM; so if you double the width you could place a fan on each half.

Even with this, seal and insulate all areas possible. Most caulking is relative easy to remove so I'd caulk the fan in and remove during the cold season. Insulating and blocking off windows that you don't mind blocking, like bathroom windows already with heavy privacy shades or ones facing the non-sunny side. There are plenty of additional, and better, ways but I can't think of any other right now like these that you can do to a rental and on a budget.