。
I'll call him first thing tomorrow morning.
Literal
Tomorrow [possessive-の] morning first him [to-に] phone let's-do.
朝一番 is a fixed expression meaning 'first thing in the morning' — 朝 (morning) + 一番 (number one / first). The volitional しよう expresses the speaker's resolution to themselves ('I'll do it') rather than an invitation. 彼に — recipient again marked with に.