I'd really recommend trying to spend the night in Galway if you can swing it, as the pubs and night life is great. If you really want to do it in a day, a good book and the train are probably your best bets.
As a note, if you decide to take one of the bus tours around the burren for the day, which many people do (and which are a great way to see a whole lot in a day), make sure you mention to the driver that you're catching the last train. I've put a few friends on them in the past, and haven't ever seen one get back at 5pm as advertised, so you'll need to make sure s/he knows to get you back to the station in time.
10

wanderingaussie
11

brodie_bruce
Yep, #10 is spot on - I took the parents on a Burren / Cliffs of Moher / Ailwee Cave tour and the tour was due back in Galway for 5:00 - we finally got back about 6:15.
A few people left the bus early and had a taxi arranged to get them back to Galway City for their train back to Dublin.
12

timothysolberg
There is a one-day rail/bus tour you could take that includes the Burren as well as Galway. No need to worry about missing your connections. See here.
Pro tip