There is a regular bus that goes several times an hour along the coast between Eastbourne and Brighton.
During the summer every other one goes via Beachy Head itself, not sure during the winter.
One of my favourite walks in the UK is to get the bus From Eastbourne to Friston Church. When you get off the bus, follow the lane down by the side of Friston Church past a National Trust Car Park and after about 15 minutes you hit the sea on top of the Seven Sister cliffs. Then simplicity itself, head east along the coast path to Eastbourne. There is a cafe at Birlin Gap and a pub at Beachy Head then descend into Eastbourne. Whole walk takes about 2,5 hours. If you want a longer walk and make a day of it stay on the bus to Exceat where the Seven Sister Country Park visitors centre is. Then follow the river path to the sea and again head east along the coastal path.
When you exit Eastbourne Station bear left from the front entrance along the High Street to the shops, the bus stops are on the right side here.