Northern Ireland can be reached by air from many cities in Britain, mainland Europe and beyond. The nearest major airports to Armagh are in Dublin (Republic of Ireland) and Belfast (two airports). Travelling to Armagh via Dublin Airport in the Republic of Ireland is likely the most convenient route, as a direct bus service exists between Dublin Airport and Armagh, unlike either of the Belfast airports. Dublin Airport is well connected with many major European airports and is ~80 miles from Armagh. Note, however, that a separate visa may be required for certain nationals. Belfast has two airports, Belfast International Airport and Belfast City Airport, both are ~40 miles from Armagh and are well connected with major UK and European airports. Getting to Armagh from either of the Belfast airports will require a bus from the airport to Belfast city centre before travelling onwards to Armagh.
From Dublin Airport
All buses from Dublin Airport to Armagh/Newry depart from Coach Zone 11 outside Terminal 1 (see map below for directions to the bus stop). The Translink X4 service connects Dublin Airport directly to Armagh, with a journey time of about 90 minutes.
The departure timetables can be seen from this link. X4 bus tickets can be booked online up to a month in advance, the booking page can be found here. Pre-booked tickets can often be cheaper than tickets purchased from the driver on the day, but must be booked at least 24 hours in advance of travel. If you wish to purchase a ticket from the driver, both Euros and Pounds Sterling will be accepted.
It is also possible to take a bus from Dublin Airport to Newry (X5 service) and then take a taxi to Armagh. This service runs every hour at xx:50, arriving in Newry 70 minutes later. Note that this route is operated by both Translink and Bus Éireann buses.
Northern Ireland Railways and Irish Rail jointly operate the Enterprise express service between Belfast and Dublin city centre, stopping at Portadown. Portadown Railway Station is ~10 miles from Armagh. The timetable can be found here. It is advisable to take a taxi from Portadown to Armagh as there are limited buses available.
From Belfast City Airport or Belfast International Airport
A bus service operates from both airports to Belfast Europa Bus Centre. The Translink 300 service from Belfast International Airport operates 24 hours a day and takes between 30 and 45 minutes to reach Belfast. The Translink 600 service from Belfast City Airport operates between 05:30 (07:30 on Sundays) and 21:45 and takes about 15 minutes. Both the 300 and 600 have a frequency of around 20-30 minutes during their hours of operation.
Translink Goldline services between Belfast Europa Buscentre and Armagh (250/251/270/271) take between 1 and 1.5 hours. Services typically operate between 07:00 and 20:00, although the frequency and operating hours of these services are reduced at weekends. Please consult the bus timetables on the Translink website, or use the Translink Journey Planner before travelling.
Alternatively, Northern Ireland Railways operate rail services between Belfast city centre and Portadown. Portadown Railway Station is ~10 miles from Armagh. It is advisable to take a taxi from Portadown to Armagh as there are limited buses available.
Returning to Dublin Airport
It is important to note that at night and on Sundays, X4 services to Dublin Airport depart from the 'Southern Regional College' bus stop across the road from the bus station. The location is indicated on the map below. A more frequent service operates between Newry and Dublin, so a taxi to Newry may be advisable if none of the X4 services are convenient for your departing flight.