Connolly attended the Abbey Grammar in Newry before commencing his studies for the priesthood in Maynooth, gaining a BA Honours degree in French before studying a theology degree at Maynooth. He was ordained in Maynooth in May 1987. He completed postgraduate doctoral studies in moral theology at the Pontifical Gregorian University, Rome, in 1991. Connolly worked as a lecturer in moral theology in Maynooth, becoming a professor in 2007. He has also lectured at St. Patrick's, Carlow College. He was executive secretary of the Irish Catholic Bishops' Conference from 1991 to 1997. He published a book on and called The Irish Penitentials in 1995 and in 2002 published the volume Sin in the "New Century Theology" series. He served as registrar at the Pontifical University Maynooth and as vice-president of St Patrick's College. On 10 August 2007, the trustees of St Patrick's College, Maynooth, announced his appointment as president of the college. He was reappointed for a second term on 29 May 2012 at the summer meeting of the college trustees. In March 2011, Connolly denied rumours that Maynooth Seminary may be closing. "Media reports today about the possible closure of the seminary are without foundation. The media outlet leading with this incorrect story was sent a reply from us yesterday and its absence from the published copy is disappointing and damaging." A year later, when the Summary of the finding of the Apostolic Visitation in Ireland was published, there was no reference to seminary closures but there were recommendations to improve the quality of seminary formation. In his capacity as rector of the national seminary, Connolly spoke frequently on the dearth of vocations to the diocesan priesthood in Ireland. Connolly also sought to raise the international profile of the Pontifical University at Maynooth and served as one of two board members for the Continent of Europe on IFCU the umbrella body for rectors of Catholic Universities worldwide. In June 2016 it was announced that Connolly was taking a sabbatical year for the final year of his statutory ten-year term as president which concluded on 31 August 2017. It was noted that the move was in accordance with the new sabbatical provisions of the Pontifical University, adopted some years earlier to reflect standard practice in the third level sector. Several months later advertisements were placed seeking a new college president, to assume office on 1 September 2017. He was succeeded in the post by Fr Michael Mullaney. In March 2017, Connolly was appointed chaplain-designate at the Irish College Paris effective 1 September 2017. He had previously served as a board member of this historic institution for ten years during its renovation and relaunch as a cultural centre and chaplaincy.
