If traveling during summer months, I've stayed at Trinity College right in the center of Dublin for a very reasonable budget. They offer student dorm rooms to travelers when school is not in session. I had a very large private room with a sink, but did have to share a bathroom with a few other rooms - but there was no wait as there were lots of toilets and showers available. The rate included free continental breakfast but you had an option to pay a few Euro more if you wanted a full hot breakfast. Great location - right on their very historic campus and the Book of Kells just steps away.
Did similar in Belfast and stayed at the dorms of Queens University. However, Queens had lots of private rooms in their dorms which had ensuite bathrooms contained within the room, so you could have your own toilet, sink and shower. They also included a full hot breakfast buffet daily in the rate. Dorms were within walking distance to Belfast Botanical Gardens, a few museums. There was plenty of taxis in the area or you could also just jump on a city bus at a stop that was just a few blocks from the dorms.
Rooms at both Queens and Trinity were very basic, but spacious and comfortable. And you get some additional benefits like free WiFi, access to basic kitchen facilities if you wanted to make a lunch or simple dinner for yourself, plus coin-op laundry.