Bottom is the complete opposite of Theseus's character in that Bottom is conceited, irrational, and allows himself to be governed by illusions. In contrast, Theseus is one of our only rational characters in the story. Theseus represents a rational and noble-minded leader of Athens.
We especially see Bottom's conceit when, in his enthusiasm, he envisions himself playing every major role in Quince's play. Bottom has especially been asked to play the male lead, Pyramus, because Bottom is apparently the best looking and most well-spoken member of their crude little company, as we learn from Quince's lines:
You can play no part but Pyramus; for Pyramus is a sweet-faced man; a proper man ... a most lovely gentleman-like man. (I.ii.77-78)
However, when Flute hesitates to accept the part of the Thisbe, female lead, Bottom gets overly excited and asks to play that part as well, thinking that he could imitate a woman's voice well. He further gets overly excited when he wants to play the part of the lion, thinking that he could roar a tremendously loud and scary roar. Since, Bottom thinks he can play all roles better than the other players, we see that he is exhibiting conceit. Not only are his desires to play all major parts conceited, they are also irrational. There is no conceivable way that he can play both a lion and the character the lion scares away at the same time. Nor is there any possible way he could play both the hero and heroine who speak on opposing sides of a wall. Thus, since Bottom has let himself get carried away by his fantasies of himself, we also see that he is not only irrational but also allows himself to be governed by illusions.
In contrast to Bottom, Theseus is quite rational. We see him acting rationally in the very first scene with respect to Egeus's request to punish Hermia should she continue to refuse to marry Demetrius. While Athenian law decrees that a father has the right to either put a disobedient daughter to death or to send her to a convent, interestingly, Egeus is only asking permission to put her to death. Theseus, being noble and rational, puts the option of sending her to a convent back into the equation when Hermia asks Theseus what could happen to her, as we see in Theseus's response, "Either to die the death, or to abjure / For ever the society of men" (I.i.67-68). Literary critic George A. Bonnard also points out that Theseus is showing his rational side when he asks to speak with Egeus and Demetrius privately. We don't really know what advice Theseus is giving them, but since Lysander has just made an excellent case that he is the better match for Hermia, according to Bonnard, we can assume that Theseus is telling them both to give up on their persistence.
Hence, we see that Bottom and Theseus are exact opposites because one is conceited and irrational while the other is noble and perfectly rational.