Kazzad13,
Birth registration only started in Ireland in 1864 and non RC marriages were only registered from 1845. Prior to that you are reliant on church records, not all of which have survived, and not all of which are on-line. There could well be other James Orr=Rebecca marriages, just that the records are either lost or not on-line yet. Monaghan to Derry is about 130 miles and, whilst perfectly possible, strikes me as a slightly unusual movement, especially prior to the arrival of the railways. If Rebecca was in Derry workhouse that generally indicated she had resided in that poor law union for some time. (Otherwise she’d have been referred back to the Poor Law Union responsible for her, eg Monaghan if she came from there). My gut feeling is she is more likely to have originated in the Londonderry area.
If her denomination was recorded as Church of England, then that’s Church of Ireland, in Ireland. I’d be inclined to check Londonderry Church of Ireland records for her baptism and for a possible parents marriage. Londonderry is in the parish of Templemore. Their baptisms start in 1642 and marriages in 1649. There’s a copy in PRONI (the public record office) in Belfast. I am not sure if they are on-line anywhere.
Orr is a common enough surname in the Londonderry area. There were 12 Orr households in that parish in the 1831 census of the county, 3 headed by people named James Orr.
http://www.census.nationalarchives.ie/search/
It doesn’t appear that the Londonderry workhouse admission records go back to the 1840s. Otherwise you might have found Rebecca’s admission records. The Board of Guardians minutes (management meetings) do exist and you might find some mention of the arrangements for Rebecca and other orphans’ voyage to Australia in them. The Guardians minutes are not on-line and a visit to PRONI in Belfast is required to view them. See this link for information on that workhouse (the main buildings are still in existence):
http://www.workhouses.org.uk/Londonderry/