I have a feeling it might rain tomorrow.

Literal

Tomorrow [topic-は] rainy-weather is-not [wonder-なかろうかと] think.

雨天ではなかろうか uses the classical negative conjectural なかろう (negative volitional of ない) — 'might it not be rainy weather?' This is a literary, somewhat archaic construction. ~と思う frames it as a personal opinion. The register is notably formal/old-fashioned.