Visa applications can be a bit confusing indeed!
The Irish Nationalisation and Immigration Service, INIS
, explain in a little more detail regarding arriving in the Republic of Ireland and then traveling up to Northern Ireland. As it depends on the passport origin, it is difficult to give the one-size covers all reply.
Most Non-EU travelers will require separate visas for the Republic of Ireland and the UK, some have a joint UK-Irish visa option, so no easy answer. Please check with the INIS website for each non-EU national and then if a UK visa is then required or a joint one exists. As the Republic is the first port of call, this is the first one to check as no entry will be granted if a visa is required here.
I hope this helps a little.