This is an interesting question because it requires some historical background to understand the irony of the spot where Caesar fell in his death throes. Remember that irony comes in several forms. In the play Julius Caesar by William Shakespeare, the irony is both situational and dramatic.
Situational irony demonstrates the difference between what is expected to happen and what actually does happen.
Dramatic irony occurs when the audience has information that the characters on the stage do not. The writer leads the audience to understand an incongruity. On the other hand, the characters do not have the information or do not connect the information they have with what has happened and are seemingly unaware of the ironic occurrence.
In the third act, the tribunes provide the information that Shakespeare gives concerning Pompey. Caesar fell at the foot of the statue of Pompey:
How many times shall Caesar bleed in sport,
That now on Pompey's basis lies along
No worthier than the dust!
The modern translation would ask the question “How many times will Caesar die and fall at the foot of Pompey’s statue to entertain the audience who is watching this play?” Of course, the Shakespeare cleverly guessed that this was an important piece of drama and that---nearly five hundred years later, it is still being performed and discussed.
In Act I, the tribunes provide the information that Shakespeare gives concerning Pompey.
Knew you not Pompey? Many a time and oft
Have you climb'd up to walls and battlements,
To towers and windows…and there have sat
The live-long day with patient expectation
To see great Pompey pass the streets of Rome.
And when you saw his chariot but appear,
Have you not made a universal shout…?
Before the civil war, Pompey was just as beloved by the people as Caesar. When Pompey fought in battles, he returned to Rome with a great reception from the citizens and the senators.
Pompey’s and Caesar’s relationship
Pompey, Caesar, and Crassus formed the first triumvirate. Crassus died. Pompey and Caesar argued over who should be the leader of the new government. This led to a civil war between the two factions. Pompey abandoned Rome, left with the government’s treasury, and was pursued by Caesar into Egypt. There the king captured and cut off the head of Pompey and gave it to Caesar as a gesture of good will. Caesar was distraught that Pompey’s body should be so desecrated. This did not stop Caesar from fighting Pompey’s sons and defeated them in Spain.
The average English citizen---poorly educated---would not have known about the relationship between the two great Romans; however, from the dialogue would realize that Caesar could have fallen anywhere in the senate room…he went down in a bloody mess at the foot of the man whom he fought and defeated in battle. That is truly ironic.