You can use a IR repeater, that would sit midway between you and the projector. Upon receiving any code from the remote control on a IR Sensor, it would simply repeat (relay) the same code (through a IR Led) towards the projector.
If you know the code to turn the projector on/off, you can have a button on the device above to send the on/off code to the projector, without the need for a remote.
Doing this is much simpler in terms of infra-structure than dealing with RS232 and long cables going through the presentation room.
For this project, even a very simple AVR (like the Atmega8, 48 or 88) can do the job. An Atmega328P is overkill.
For the parts you'd need:
a IR receiver, like Vishay TSOP1738
a IR Led (you'd have to find out if your existing remote uses narrow or wide beam)
a few resistors and capacitors
I don't think all the parts above would cost more than US$ 5 + an Arduino.