MOSFET seems to be the best solution. No need for PWM if you just want on/off control.
This:
Can be interpreted 2 way, either or both could be relevant.
- You could mean that the time from some event that triggers turning the LED on or off must not be more then 1μs
- You could mean that the rise and fall time, as in the time it takes to apply or remove the power, cannot be more than 1μs
Those are different requirements, which one applies?
Knowing what you are really trying to do would get you better answers.