Probably the cheapest and easiest way to get to Limerick is flying to Dublin. Being the biggest airport in Ireland, many low-cost airlines, such as AerLingus and Ryanair, regularly offer deals from various European cities to Dublin, usually starting around 16€ one-way. When you arrive in Dublin, you can take a bus directly from the airport. You can choose between the Dublin Coach, a bus company that takes you to Red Cow bus stop of the Luas (tramline), at this bus stop you can change onto a bus to Limerick that takes you straight to the University of Limerick campus. Tickets are available with the bus driver or online. You can buy the two fares seperately or together, whichever you prefer, I usually tend to get them both together. They cost 17€ (or €15 if you book online) and the duration of the trip including the transfer is about 2:45 hours. Another option is to take the City Link bus. It's a direct fare, costs 25€ and only takes 2:25 hours, but it does not stop directly at UL, but at a pub called Hurlers, about 10 minutes walk from university.
Another option would be to fly into Shannon Airport, with approx. 30 minutes drive, the closest airport to Limerick. UL offers a pick-up service to students, but before you arrive, you will definitely receive an email from the International Office about this.