In Shakespeare's Julius Caesar, who should have ruled Rome between Brutus, Cassius, and Antony?

This would have to be a matter of opinion based on Shakespeare's characterizations of the three men, which were based on his reading of Plutarch's Lives of the Noble Greeks and Romans, written over a hundred years after the events depicted in Julius Caesar. Brutus seems to be the man who would have been the best ruler because of his high moral character. He was highly esteemed by everyone, including by Julius Caesar himself. Mark Antony actually did become one of the rulers of Rome, along with Octavius. No doubt Antony had many of the qualifications of a good ruler, including his extensive military experience and sound judgment. However, as Shakespeare shows in his Antony and Cleopatra, Antony was notoriously hedonistic and extravagant. He was quite different from Cassius, who was sober, serious, and conservative. But Cassius was also selfish and dishonest. He would have been a corrupt ruler. When he and Brutus are having their violent quarrel in Act 4, Scene 3 of Julius Caesar, Brutus says:

Let me tell you, Cassius, you yourself
Are much condemn'd to have an itching palm,
To sell and mart your offices for gold
To undeservers.

This suggests the kind of ruler Cassius would be. He would be likely to make himself king and possibly emperor; whereas Brutus would try his best to have a totally honest and open government. Brutus seems the best choice for ruler, Antony the second-best, and Cassius the worst. If Brutus and Cassius had managed to beat Antony and Octavius at the battle of Philippi and the victors had become joint rulers of Rome, they would have been continually at odds. Cassius might have become sole ruler, because he would have been likely to have Brutus assassinated. On the other hand, Brutus might have used his popularity and high moral authority to expel Cassius from their partnership. Brutus is a nice guy, but he has a temper. When he and Cassius are quarreling in Act 4, Scene 3, Brutus actually threatens Cassius with chastisement.

CASSIUS
I an itching palm?
You know that you are Brutus that speaks this,
Or, by the gods, this speech were else your last.

BRUTUS
The name of Cassius honors this corruption,
And chastisement doth therefore hide his head.