Obviously the character of Caesar is more famed for his pride than for anything else, and this is a key feature of his character. Examples of fear are somewhat harder to come by, but have a look at Act II scene 2 for examples of both.

This scene is where Calphurnia tries to persuade her husband not to leave the house today because she has had a dream warning her of Caesar's death if he does. Look how Caesar answers her fears after news from the augurers:

Danger knows full well

That Caesar is more dangerous that he.

We are two lions littered in one day,

And I the elder and more terrible.

And Caesar shall go forth.

Here Caesar compares himself to "Danger" and says he is more potent and more "terrible" than Danger as a force.

I guess you could say the only fear Caesar displays is a fear of a loss of prestige, status or honour. It is this that allows Decius to convince him so easily to go to the Senate. Ater Decius' speech where he says Caesar might not get the crown after all if he does not leave the house, Caesar replies:

How foolish do your fears seem now, Calphurnia!

I am ashamed I did yield to them.

Give me my robe, for I will go.

It is fear of losing more power and prestige that makes Caesar bow to the rhetoric and persuasion of Decius, and ignore the premonition of his wife.