Pat, your prospects of finding YOUR Patrick Burke from such scant information are very poor. Even the VRI (which is not noted for its comprehensiveness) contains two possibilities:
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BURKE, Patritius Christening
Gender: Male
Christening Date: 25 Jan 1829 Recorded in: Athlone and Drum, Roscommon, Ireland
Collection: Roman Catholic
Father: Jacobi BURKE
Mother: Mariae CROGHAN
Source: FHL Film 989750 Dates: 1826 - 1832
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BURKE, Patritium Christening
Gender: Male
Christening Date: 11 Mar 1831 Recorded in: Lisanuffy, Roscommon, Ireland
Collection: Roman Catholic
Father: Edvardi BURKE
Mother: Honoriae MCCORMICK
Source: FHL Film 989745 Dates: 1831 - 1833
++++++++++++++++++++++++++++++++++++++++++There will be many others, whose records would have to be dug out of parish records. To find the right parish, you will need to find something that POINTS to that parish. Burkes are fairly ubiquitous around Roscommon, so you need something else. Are you aware of any other family that migrated with your ancestor? Do you know Patrick's mother's maiden name? When you have at least a couple of surnames to work with, you can sometimes use Griffiths and Tithe references to find common parish, and even townland, locations for the various families.
brianfw@ozemail.com.au