In "Julius Caesar", how important are ideas of being "honourable" and "noble" in the play?
Very. Part of the society that Shakespeare depicts is one in which men are obsessed with being thought of as honourable and noble - sometimes more so than being honourable and noble.
Cassius is the prime manipulator of this flaw - and, you guessed it, it's part of Brutus' arrogance that he's set on his own ideas of how he is noble and honorable. So Cassius says to him
Age, thou art shamed!
Rome, thou hast lost the breed of noble bloods!
History is shamed, because Rome no longer comes up with honourable people. And, as Cassius says to himself at the end of the scene, Brutus' idea of honour is precisely the way to attack him:
Well, Brutus, thou art noble; yet, I see,
Thy honourable mettle may be wrought
From that it is disposed...
Cassius presents the conspiracy, continually, as honourable and noble - a worthy cause:
I have moved already
Some certain of the noblest-minded Romans
To undergo with me an enterprise
Of honourable-dangerous consequence...
These are my italics. But you get my point. And later in the play, Antony uses ideas of honour to turn the tide against the conspiracy:
For Brutus is an honourable man;
So are they all, all honourable men—
And at the end, Antony's partially ironic (fully ironic? Who knows) eulogy over Brutus complements him based on the ideals of honour:
This was the noblest Roman of them all.
You could analyse all this much further and in more detail. But it is very, very important to the play.