We arrived last April and did not apply for ARC in 8 days. We went to the immigration centre and was told not to bother until me and my partner were both working. Cyprus is in the EU so it is not mandatory to apply if you don't wish to
As it happens we have our yellow slip now and this was done in November lady year, 7 months after we arrived.
The immigration centre is on the way to Dherynia opposite the fire station. It's at the back of the new social centre.