There are Ardrys/Ardreys in various Co. Tyrone marriage records 1845+ in and around the Dungannon/Donaghmore area in Presbyterian and Church of Ireland churches, but I don't see a "Robert" in on-line civil registration indexes.
I assume you found Robert's 1813 christening in Protestant records, which for Donaghmore would probably be St. Michael's Church of Ireland (Donaghmore parish) at Castlecaulfield, with records back to 1748. The LDS church has microfilmed many of those church records, and indexes are on the FamilySearch web site. I see that William and Jane Ardry christened other children besides Robert: a daughter Jane in 1816.
Are you interested in finding more on the whole Ardry family, William and his family and ancestors, or just the individual son Robert? There's helpful Irish ancestor information on-line, if you're researching all the family to identify your ancestors.
I see in other queries that you're looking for information on "William Ardrey born about 1782 who immigrated to Lewiston NY. He married a Rosanna (last name ?). Any information on their marriage, immigration or children would be appreciated." Also: "great,great,great grandfather - William Ardrey. He lived in Lewiston from approximately 1820 until his death in 1869. His first wife was Jane and he subsequently remarried a Rosanna Ardrey. I am looking for his obituary or cemetery where he is buried."
I can't suggest research lines in New York, but here are a few notes on Ardreys/Ardrys in Ireland. If William was born in the 1780s, then you'd want to find anything you could about Ardreys that early (at least).
Pre-1858 Wills and Administrations indexed at PRONI (www.proni.gov.uk) show that John Ardry of Mullineal, Co. Tyrone, had a will probated in 1783. Also known as Mullyneill, this townland is in Aghaloo parish, northeast of Minterburn and northwest of Dyan.
William Ardry of Dralskey(?), Co. Tyrone, left a will in 1818. Neither of these original wills exist today, only the indexes of names and dates has survived.
Robert Ardry of Mullybrannon, Clonfeacle parish, was counted as a Freeholder (man holding enough property to vote) in 1797. He signed at Dungannon, just a short distance from Donaghmore. There are quite a few records on Ardrys of Mullybrannon/Mullybrennan, including wills and deeds.
There don't appear to be any Ardrys in Donaghmore parish (where you said Robert Ardry was christened) at the time of the Tithes (1820s-1830s), but by the time of Griffith's Valuation (1859-1860), "John Ardry" was counted as a landholder at Mulnahunch townland, Killeeshill parish, Co. Tyrone. The Killeeshil & Clonaneese Historical Society is a good place to search for records in that area: http://killeeshilclonaneese.orghttp://killeeshilclonaneese.org. And the Donaghmore Historical Society is a good place for research in and around Donaghmore: www.donaghmorelivinghistory.com/dhs.htm
There were more Ardrys in Co. Down than Co. Tyrone. For instance, in 1865, a "Robert Ardery" lived at Ballyvally, Co. Down. His will was probated in 1867 (see PRONI indexes, ref. # D1046/127). This may not have been your Robert, but it's worth checking out.
Colby Ardrey of Columbia, Missouri was researching the Ardrey family of Co. Tyrone and Co. Down in 1999. You can do a Google search for his/her messages, which include this ancestor data: "I am researching the Ardrey/Ardery/Ardry family who are supposed to have been from County Tyrone and County Down. My g-g-g-g-grandfather, Alexander Ardrey immigrated from County Tyrone around 1807 and finally settled in Scott County, Indiana around 1820. He was married to Susannah, maiden name unknown. I do not know for sure if they were married in Ireland or US. Any information would be greatly appreciated!" Comparing family data and genealogy findings with this researcher might be helpful.
Lower Clonaneese Presbyterian Church registers (Co. Tyrone) include Ardreys of Cranslough in the 1890s through early 1900s-- see www.cotyroneireland.com/churchrecord/clonaneeselowerbaptisms.html. These families were counted in the 1901 and 1911 censuses, and there might be people of the name still living in that area today. Getting in touch with them might be helpful.