Running LED on low voltage for a long time
You MUST specify your battery "chemistry" as "2.4V" could mean several things and results will vary. That sounds like 2 x AAA NimH cells. Are you limited to that capacity or the size chosen or ...? 

I'll assume 2.2V = 2 x 1.1V = 2 x average output voltage of a NimH cell at moderate load.  

If using NimH then voltage will fall to 2V at end of battery life so system must run off 2V.

At say 2.2V x 1000 mAh = 2200 mWh battery energy and say 80% converter efficiency, for 4 hours operation you can get  2200 mWh x 80% /4 hours = 450 mW.

This is so close to 1/2 Watt as to not matter. You can get 80% "easily enough" - 
90% may be possible but is harder across the whole voltage range. 

A very common and low cost IC is the LM319 / LM339  dual/quad comparator.  These are almost certainly easily obtained where you are.   

These will operate with Vcc from 2V to 36V and can be used to make oscillators and to act as switching elements in switching reqgulators. They do not have enough current drive to drive the LED directly but will drive a suitable MOSFET or bipolar transistor plus inductor in a boost converter circuit. 

LM319 dual comparator data sheet  

LM339 quad comparator data sheet    



The circuit below, which is Fig 3 on this page could have the 555 oscillator replaced with an LM339 oscillator and a current sense resistor between LED cathode and ground would provide LED-current feedback control. Components could be adapted to suit.