At one time(long ago) there were five provinces in Ireland, 'Meath' being one of them - it was later divided into the present counties of Meath and Westmeath. There is no "eastmeath" as such, but it would refer to the coastal region (which is not very much, in Co. Meath), so it would depend on how far inland one wished to go ! Oldcastle is in the northern part of the county, well away from the coastal region...... 40+ miles from the sea.
General Register Office is Lombard St, Dublin 2, but there is now a 'decentralised' office, in Roscommon, for postal applications. Look up GRO, and there is a website.
MikeW.