Melissa,
Statutory birth registration only started in Ireland in 1864 so you won’t find a birth certificate for Mary. You might find her baptism if you know her denomination and if the relevant set of records still exist.
You haven’t said what denomination Mary was. However tradition was to marry in the bride’s church, and so that can be a clue. Mary and Thomas married in Malone Presbyterian so there’s a fair chance she was Presbyterian. If she was born around 1829 in Drumbo then I would search Drumbo Presbyterian’s records. They start in 1699. There are gaps but there are records for around 1829 so you might be OK. The records are not on-line anywhere so far as I am aware but there is a copy in PRONI (the public record office) in Belfast. A personal visit is required to look them up, or you can ask PRONI if they will do it for you (for a fee) or you could employ a local researcher. Researchers in the PRONI area: http://sgni.net
You asked about possible siblings for Mary. If Drumbo Presbyterian was the family church then you should get siblings details from their records. In addition, Rootsireland (subscription site) has a search function which allows you to search marriages under surname and father’s name. I did that for Cochrane plus father William, in Counties Antrim & Down for the years 1845 - 1865. I found 39 matches. Too many for me to go through but you might want to do that. Bear in mind that, according to the 1851 marriage certificate, William was a spoonmaker. That’s a fairly unusual occupation and might help. Mary was living in Dunmurry at the time of her marriage but it doesn’t mean that her siblings or parents were though. Dunmurry is quite close to Drumbo.
I checked Griffiths Valuation for 1861. There were 4 William Cochranes listed in Co. Down. None in Drumbo though. Your William may have been dead or in a house of too low a value to be listed.
