"Murder in the Phoenix park" refers to the assassination, May 6th 1882, of Lord Frederick Cavendish, British secretary for Ireland (shown here), and Thomas Henry Burke, his undersecretary. They were stabbed to death in Phoenix Park by members of the Irish National Invincibles, a splinter republican group. Note that Myles Crawford gets the year wrong. Stephen was born in 1882, presumably in February, so indeed before the Phoenix Park murders.
Cavendish and Burke were #2 and #3 in rank in the British Dublin Castle government of Ireland. They were on their way to meet #1 the Lord Lieutenant Earl Spencer (John Poyntz Spencer, 1835 - 1910, shown in this photo) in the viceregal lodge. Thirteen men were later arrested and charged with conspiracy to murder. They were apparently out only for Burke as a "Castle Catholic." Two (James Carey, and Michael Kavanagh) turned state's evidence, five (Joe Brady, Dan Curley, Michael Fagan, Thomas Caffrey and Tim Kelly) were hanged, and 3 were sentenced to penal servitude. In the aftermath, Irish nationalist leader Charles Stewart Parnell denounced 'these vile murders' and offered to resign from parliament, an offer turned down by British Prime Minister Gladstone; this increased Parnell's already huge popularity in both Britain and Ireland, being seen as a moderate reformer who disapproved of violent or terrorist tactics. However, Gladstone's Minister Lord Hartington, the elder brother of Lord Cavendish, split with Gladstone on Home Rule in 1886 and 1893 and led the breakaway Liberal Unionist Association which allied itself to Lord Salisbury's conservative governments.
"He pushed past them to the files.
- Look at here, he said, turning. The New York World cabled for a special. Remember that time?
Professor MacHugh nodded.
- New York World, the editor said, excitedly pushing back his straw hat. Where it took place." (U7.634)
"Tim Kelly," (U7.639)
"or Kavanagh I mean," (U7.639)
"Joe Brady and the rest of them." (U7.639)
"Where Skin-the-Goat drove the car. Whole route, see?
- Skin-the-Goat, Mr O'Madden Burke said. Fitzharris." (U7.640)
"He has that cabman's shelter, they say, down there at Butt bridge. Holohan told me. You know Holohan?
— Hop and carry one, is it? Myles Crawford said." (U7.641)

Butt Bridge is a road bridge across the Liffey that joins George's Quay to Beresford Place. The original bridge was a structural steel swivel bridge, opened in 1879. It was named for the leader of the Home Rule Movement Isaac Butt (shown on this CDV), who died that year. The swing section, made of wrought iron and weighing 200 tons, ran on a series of cast spoke wheels and was powered by a steam engine, which was housed on a timber pier on the downstream side of the bridge. The swing action allowed boats to pass and berth in the river as far upstream as Carlisle (later O'Connell) Bridge.
"— And poor Gumley is down there too, so he told me, minding stones for the corporation. A night watchman
Stephen turned in surprise.
— Gumley? he said. You don't say so? A friend of my father's, is it?
— Never mind Gumley, Myles Crawford cried angrily. Let Gumley mind the stones, see they don't run away. Look at here. What did Ignatius Gallaher do? I'll tell you. Inspiration of genius. Cabled right away. Have you Weekly Freeman of 17 March? Right. Have you got that?
He flung back pages of the files and stuck his finger on a point." (U7.645)
"- Take page four, advertisement for Bransome's coffee let us say. Have you got that? Right.
The telephone whirred.

- I'll answer it, the professor said going.
- B is parkgate. Good.
His finger leaped and struck point after point, vibrating.
- T is viceregal lodge. C is where murder took place. K is Knockmaroon gate." (U7.654)

The coffee was Branson's, as discovered by Harald Beck.
This ad suggested by Eamonn Finn. Image Evan 6168 Copyright British Library Board. All Rights Reserved.
The Viceregal Lodge, located in Phoenix Park, was designed by park ranger and amateur architect Nathaniel Clements in the mid 18c.; later, architect Francis Johnson (who designed the GPO) added the porticoes. The lodge was bought by the administration of the British Lord Lieutenant of Ireland (= Viceroy) in the 1780s.
The Viceregal Lodge became, from the 1820s onwards, the 'out of season' residence of the Lord Lieutenant, where he lived for most of the year. During the 'social season' (January to St. Patrick's Day in March) the Viceroy lived 'in state', in the Viceregal Apartments in Dublin Castle.
"The loose flesh of his neck shook like a cock's wattles. An illstarched dicky jutted up and with a rude gesture he thrust it back into his waistcoat.
- Hello? Evening Telegraph here. Hello...? Who's there...? Yes... Yes... Yes." (U7.663)
"- F to P is the route Skin-the-Goat drove the car for an alibi. Inchicore, Roundtown, Windy Arbour, Palmerston Park, Ranelagh. F.A.B.P. Got that? X is Davy's publichouse in upper Leeson street." (U7.667)
"The professor came to the inner door.
- Bloom is at the telephone, he said.
- Tell him go to hell, the editor said promptly. X is Burke's publichouse, see?

- Clever, Lenehan said. Very.
- Gave it to them on a hot plate, Myles Crawford said, the whole bloody history." (U7.670)
"- History! Myles Crawford cried. The Old Woman of Prince's street was there first. There was weeping and gnashing of teeth over that. Out of an advertisement. Gregor Grey made the design for it. That gave him the leg up." (U7.684)

This PC shows Prince street between the Hotel Metropole and the GPO. The Old Woman = The Freeman's Journal, located just behind the Metropole.