Oh, according to the internet not very long!
Well, it depends. If you have a heavily planted tank it can make a big difference, as the plants (and substrate) will harbour plenty of beneficial bacteria and the plants themselves will break down toxins. If you've no plants then the water will start getting poisonous pretty quick.
but I need to figure out how to run an air pump with a battery or something.
An air pump will make no difference - your fish won't run out of oxygen, they will be slowly poisoned. The filter works by building up colonies of bacteria that break down ammonia and nitrites into harmless nitrates. If the tank water isn't circulating over these filter colonies then the ammonia and nitrate levels will increase to levels that will kill your fish.
First thing you should do is
stop feeding your fish. Every bit of food you put in will end up breaking down into toxins. Tropical fish can survive long periods without food - four days will be no problem.
The next thing you should do is a large water change every day - at least 50%, probably more. If the toxins aren't being removed by the filter, then you need to mechanically remove them - water changes are the only way to do this.
The other thing you should keep in mind is that while the filter isn't running, the bacterial colonies will start dying, so even when you turn it back on it may be working way below its usual level. So I'd carry on with not feeding and big water changes to be safe.
Anyway, this is what I'd do. I've kept tropical fish for a long time, but I'm not an expert, so do some research. Just be careful, as there is an awful lot of terrible, out of date, rubbish out there.