will a 48V DC motor work with a 36V Battery and controller
I take it you don't have any specifications for the motor showing the acceptable voltage range?

Absent the spec, I would be nearly certain that the motor itself would perform just fine on 36V.

For the motor controller; the ones I'm familar with for these kind of applications are actually current-limited, not power-limited.
So, your 36V 1kW controller will likely be a 28A controller. This means your 48V motor will not be 1000W on that controller, it will only be 750W.
That means that while the controller won't be overloaded the maximum power output might not be sufficient for your needs.

Speed for a brushed DC motor is related to voltage too, so your motor will only be able to turn at 3/4 of its rated speed.

There are a lot of assumptions here, so it would best to try and verify as many of them as possible by getting detailed specs for the components (or asking the vendors).