1845 in the Uk, particularly Ireland, was a year of heightened Protestant versus Roman Catholic antagonism. This is because the government of the UK, with Sir Robert Peel as Prime Minister, had proposed an increase in government funding to St Patrick's Roman Catholic Seminary at Maynooth, near Dublin.
Since 1795 this had been funded by the government to train Roman Catholic priests. Dublin University already trained Protestant ones for ordination in the Church of Ireland. By 1845 Maynooth was in need of significant repairs. Sir Robert proposed to increase the annual grant from £9,000 to £26,000, and to provide a further one-off sum of £30,000 for repairs etc.
This provoked outrage and opposition amongst many Protestants in all parts of the UK and anti-Maynooth conferences, called Great Protestant Meetings, were held in London in May, and in Dublin in June. A petition of over a million signatures opposed it.
The earlier 1840s saw a campaign for the Repeal of the Union, that is for Ireland and Great Britain to have their own separate parliaments again. This was also a divisive issue, being seen by Protestants as a threat. Sir Robert's Maynooth proposal was intended to convince people that the government was fair to Irish Roman Catholics, and so reduce demands for Repeal of the Union.
It is against this background that, on the evening of Tuesday, 1st July, 1845, Mr Short emerged from the Rotunda Gardens wearing an Orange lily, and subsequently got into a disorderly fight with a 14-year-old boy, Master O'Brien. Neither young gentleman seems to have been seriously hurt. Both were charged with being disorderly and fighting, but neither was charged with inflicting actual bodily harm.
What was Mr Short doing in the Rotunda in the first place, and why was he wearing an Orange lily? The date, July 1st, is a clue. That the newspaper article quoted by OP refers to an Orange lily, rather than an orange lily, is another clue that the flower was, in the context, understood to be a badge or symbol of the Orangemen.
The Rotunda is the Dublin equivalent of the Assembly Halls or Rooms found in many other cities. It is a collection of meeting rooms of various sizes.
The Batle of the Boyne took place on July 1st, 1690, in the Old Style Calendar, equivalent to July 12th in the New Style calendar. The final battle of the Williamite campaign was at Aughrim on July 12th, 1691 (Old Style). Nowadays, the 12th July is celebrated as the Anniversary of the Boyne with colourful parades in Northern Ireland, Merseyside and elsewhere; and orange lilies still feature in them.
In 1840s Dublin, the original Boyne date of July 1st was kept by the city's Orangemen. I haven't found any report of this in 1845, my access to old newspapers is limited without suscriptions. The Spectator in 1844 refers to a meeting of Dublin Orangemen in the Rotunda on July 1st. It says the oratory was of a violent description and that 2,000 persons sat down to tea. It also says banners and badges were rife.
So, it is extremely likely that a similar Orange tea took place on 1st July, 1845, in the Dublin Rotunda. This is referred to, perhaps ironically, in the article cited by OP as a "Great Protestant Meeting".
So, when Mr Short left the Rotunda, and emerged fom the Rotunda Gardens around it, he had just attended an Orange celebration and this was clear to see by the fact he was still sporting his Orange lily. In the heightened tension of the time, especially, this annoyed some of the crowd. These had, quite possibly, gathered there to express their opposition to the Orange event.
Quite what happened next is not clear. Mr Short cannot have been violently attacked by a mob, or he would have been seriously injured. Perhaps the boy attempted to grab the lily. The police, who must have been aware of the situation, seem to have intervened very quickly to arrest both of them, and so prevent any widespread disorder.
This, I hope, gives some context to why Mr Short emerged from the Rotunda Gardens, on July 1st, 1845, sporting an Orange lily, and why this might have led to some kind of fight between him and 14-year-old Master O'Brien.