There was a time when you can mine ETH with rx480 and 580 with 4GB memory using a particular miner, i'm guessing it's lolminer? Now that the DAG file is very close to 4.9GB I doubt this trick still works. Anyone using this strategy with 4GB cards?.
You can mine ethereum on 4gb graphics cards, but your hashrate will be less than 1 megahash. That's how it was when I last tried with a DAG file size of about 4.2gb. Mining may not work now, but you will not be mining ethereum with this hash rate, even if you can start mining.