I suggest the following:
A digitised copy of Henry Munro's Will (dated 16 Jun 1864) is available online from the PRONI Will Calendars website (see http://apps.proni.gov.uk/DCAL_PRONI_WillsCalendar_IE/WillsSearch.aspx
) - he died on 22 Jun 1864 and Probate was awarded on 1 Feb 1865 at Belfast to his widow Maria Munro and Robert Smith, a Merchant, both of Warrenpoint, the Executors of his estate. (NB you need to search on MUNRO as the surname). He left bequests to his son William and three daughters, including a property in Duke Street, Warrenpoint.
Others in the family can be found by using MONRO as the surname. Of particular note should be Sarah Jane Monro - William's sister, died 1897, and Robert Hector Monro (son of John who died in 1901) is also available on the same site, (this is the Robert H Munro in the 1901 Census put forward by StLouisGuy). Also John Monro of Warrenpoint who died in 1894 leaving his estate to his widow Elizabeth - check GRONI for a marriage between John Monro and Elizabeth Monro - if found this may be another of William's sisters.
If you go to the equivalent Will Calendars site for Dublin online (see http://www.willcalendars.nationalarchives.ie/search/cwa/results.jsp?deceasedSurname=monro&deceasedForename=&beneficiarySurname=&beneficiaryForename=&placeDeath=&probateOrAdministration=&dateDay=&dateMonth=&dateYear=&search=Search&sort=&pageSize=&pager.offset=0
) and look through all the records for Monros in County Down this should allow you to build up a picture of the family members.
In the Griffiths Valuation from the early 1860s Henry and John Monro each have a property in Duke Street. This leads me to a working assumption that Henry and John were certainly related and possibly brothers.
In terms of paid online sites, the General Records Office of Northern Ireland (GRONI - see https://geni.nidirect.gov.uk/
) website holds records of Birth (1864- ), Marriage (1845- ) and Death Certificates (1864- ). You can search on a forename+surname combination over a year range of up to 4 years for free, however this just returns the number of hits and isn't really useful. If you buy some credits, however you can search on and have displayed a digitised version of each Certificate (tip: if displaying a Certificate choose the Full version - 5 credits). For Birth Certificates you get name, birthdate, father's name, mother's forename and maiden name. For Marriage Certificates you get bride and groom names, occupations, marriage venue, father's names and occupations, and finally for Death Certificates you get death date, cause of death and who was present at death.
To go back further in time you need to go to one of the paid genealogy sites - my preferred one is RootsIreland (see https://rootsireland.ie/ifhf/subscribe.php
). Subscription prices start with a pretty reasonable £8 for 24 hours access. It may be pushing it a bit to get Henry and Maria's Marriage Certificate via this site, however you may get Birth/Baptism and Marriage records for the children (Tip: when searching for marriage records for Henry or Hector's children, enter Monro (or Munro) as the surname, and Hector/Henry as the father's firstname, then choose a year and choose a 10-year time period. Use a similar approach when searching for Births/Baptisms - you can also add the mother's forename for Births/Baptisms however bear in mind this wasn't always recorded)