Is Romeo believable in his soliloquy?

Assuming you mean his balcony scene soliloquy in which he compares Juliet to the sun, yes, Romeo is completely believable. He has just fallen passionately in love. He is young, heady, without restraint. His over-the-top comparison of Juliet to the sun, his extravagant claim that her eyes are so bright that if they were stars in the night sky the birds would think it was daylight and start to sing (you might listen to an old Carpenter's song called "Close to You" to catch the gist of Romeo's mood), his adoration of her in all ways, is completely consistent with how people, especially young people, feel when they fall giddily in love. At first, the beloved is all perfection, a god or goddess, the miracle that will complete us. Shakespeare captures this overflowing exuberance in Romeo's exaggerated language. Whether the glow of first love is an illusion is another question--Juliet undoubtedly is not all perfection-- but that people feel it as strongly as Romeo did is entirely believable.