朝一 (あさいち) literally 'morning-first,' a casual term for 'first thing in the morning.' 予約が入る ('a reservation comes in') is the standard phrasing for bookings from the service provider's side, using が to mark the new information. ~んだが is the explanatory んだ plus the trailing が contrastive — 'there's this, but...' setting up the offer. 試しに is the adverbial 'as a trial, for testing purposes.' やってみるか uses the ~てみる ('try doing') auxiliary plus か (casual invitation). A natural, friendly masculine offer.