Most LEDs have rise and fall times below 30 ns, usually below 20 ns. Similarly, fairly easy to find a power MOSFET that can switch in 20ns or even faster. Hence, you requirements are easy.
Constant current driver may pose a problem - the inductor slows down its response time. Some special design is required.
Although not perfectly optimal, the overall easiest solution, by far, is to use a bog standard regulated voltage supply and a plain old series resistor - the only downside is, to maintain some current regulation, you need to drop some meaningful voltage over the resistor, let's say at least 20-30% of the total supply voltage. So it's not as efficient as a constant current drive, but can be acceptably efficient. For even faster switching, you can use a small bypass capacitor parallel to the series resistor, but this shouldn't make a difference with your fairly slow pulses. Even without the small capacitor, you are already at full LED power after about 20-30ns, which is a small percentage of your 500ns pulse.
Do beware of stray inductance. Keep the physical loop as small as humanly possible. If you have a lot of LEDs, make the LED series layout a tight, thin "U shape" so that the opposing currents go as close to each other as possible, and Vdd and GND are right next to each other (preferably sandwitched on a multilayer PCB):
VDD-Res-LED-LED-LED-LED-|
GND-FET-LED-LED-LED-LED-|
Spray ceramic capacitors (snubbed with a larger electrolytic) between your Vdd and GND.
If your duty cycles are high, you'll have two opposing constraints, thermal design would benefit from loose layout, while you need tight layout for low inductance.