There is a direct bus service from Belfast to Ballywalter. It goes from Belfast Laganside Bus station. Note that this bus station is not the main bus station in Belfast (which is at the Europa Hotel/Gt Victoria St) but is located by the Waterside Concert Hall on the River Lagan. Still in the city centre but about a quarter of a mile away.
Route number 7 goes to Ballywalter. Typically it takes 1 hour 25 minutes. On weekdays there are departures at 10.35, 11.35 & 12.35. Returns at 1200,1300,1400 and 1705. (There are other services). You can view the full timetable on this link:
http://www.translink.co.uk/Routes-and-Timetables/all-buses/?lrn=&spn=Ballywalter%2C%20Main%20Street&spns=10002546
Get a day return fare from the driver (normally cheaper than two singles). There are no concessionary fares for being over 60. (Well there are, but you have to be resident to qualify. Public transport is free if you are over 60, and resident in NI, but you need to have a concession card to benefit). The graveyard is on the Dunover Rd which is on the north side of the village. Ask the driver to drop you there. (The graveyard looks to be a walk of about 300 or 400 yards from the main coast road).
Other than the bus, you would need to either take a taxi (rather expensive I would think) or hire a car. No railway goes near Ballywalter.
Here’s a link to a site where someone has photographed all the gravestones. The one you are looking for is there, and is on this link which also includes a map of the area:
http://www.yorick.homecall.co.uk/Whitechurch/FrameSet.htm
