If by, 'is it possible?,' you mean could an item take just one day from the UK to Dublin, the answer is probably 'Yes'. If you posted an item at, say, Liverpool sorting office in the early hours of Monday morning, I would think that it might be quite likely to be delivered in Dublin at some time on Tuesday. (I chose Liverpool for my example because most mail for Ireland probably goes through John Lennon Airport, which is used for large amounts of air mail. The further away from Liverpool, or possibly London, you post an item, the less likely it would be to get to Dublin by the following day).
However, if what you really want to know is whether there is any Royal Mail service which offers a guarantee of next day delivery in Dublin, the answer has to be 'No'. I've got a copy of the Royal Mail's publication 'Mail Made Easy' in front of me. Page 9 has maps showing the 'guaranteed' delivery times which are available for different areas. It specifically states, "Republic of Ireland: No Special Delivery service available".
Chris