Seven Sisters in Winter?

There are certainly trains to both Brighton and Eastbourne most of the time. There are a couple of buses connecting the two, but the 7 Sisters are much closer to Eastbourne. I'm at work so I don't remember the number. There's not much to see from even the top deck but there is a stop from where you can walk across and over to the tops of the cliffs, and another path down to the sea.

The 7 Sisters are open every day but the trains to take you south won't be. There is virtually no rail service anywhere in the country on Christmas Day and Boxing Day, and although there are services as far south as Gatwick (from Victoria) on Boxing Day they are diverted to East Grinstead from whence it is a bus to the airport.

There will be virtually no public transport of any kind anywhere in London on Christmas Day and a depleted Sunday service on buses and tubes (but still virtually no trains) on Boxing Day.

You should know that the Monday after Christmas has been declared a Bank Holiday this year.

Regarding the cold temperatures you worry about there haven't been any on the south coast yet this year. It has been very mild, extremely windy - with 4 named storms so far this year and the last, Desmond was a real doozie - and very wet.

Depending on if there is any sun - unlikely so far this month - sunrise is around 8-ish and and sunset just before 4, with it quite dark on many days well before that.

Take a train to Eastbourne, which is 5.5 miles from Birling Gap. Once at Eastbourne station take bus 13X which runs hourly from Eastbourne Terminus Road to Brighton via Birling Gap.
Operates Sundays & bank holidays all year. Weekends from Easter to September. Daily from June to September.
http://www.buses.co.uk/travel/service.aspx?serviceid=1233

In July 2011 we had a Beachy Head Tour booked with Sussex Coastal Voyages ... http://www.sussexvoyages.co.uk/beachyhead.html ... but bad weather forced cancellation. So, even in mid summer, the weather can play havoc.
We drove the cliff top road and stopped for a short walk and some photos in a buffeting wind.
There is a wonderful pub, The Tiger Inn, on the gorgeous village green at East Dean.

http://www.beachyhead.org.uk/
http://www.nationaltrust.org.uk/birling-gap-and-the-seven-sisters

You could bus to East Dean from Eastbourne, have lunch at the Tiger Inn, walk to Birling Gap, then continue along the cliff tops to Cuckmere Haven and finally into Exceat, from where you can take the bus again all the way to Brighton. There is the option to walk along the beach under the Seven Sisters if you are feeling adventurous, but you would need the tides (and weather) to be in your favour as well having the correct clothing and footwear for the stone beach.