Methodism is a splinter from the Church of England/Ireland. It didn’t start as a separate denomination in Ireland until 1816. Even after that date many Methodists continued to use the Church of Ireland for baptisms and marriages, in some cases right up to the 1870s. So you don’t get any early Methodist records in Ireland. The earliest are probably from around 1830/1840. So my advice prior to that would be to search Church of Ireland records for early baptisms and marriages.
I typed Robinson and Dungannon into the local phone book and found 60 listed, so plenty to choose from. A problem you will face is that most people in Ireland don’t know their own family tree (and often aren’t all that interested) so it can be tricky finding who the living descendants of your family are. They don’t know themselves.
UK phone book: http://www.ukphonebook.com/?cd=tels&uen=GQ9aqxUMYftSv4ku...
You could try writing to the pastors of the Methodist churches in that area, enclosing an open letter to any Robinson in the congregation, inviting them to get in touch. Here’s a site that lists all their churches:http://www.irishmethodist.org/find-a-church/index.php
Methodists in Ireland generally don’t have their own graveyards and so use other denominations eg the Church of Ireland, so that could account for the burial in Drumcree. The Church of Ireland does keep burial records but not for other denominations burials. However the custodian of the graveyard may have other information eg who is buried in a given grave.