"Αρχίζω να πεινάω" is the correct form. (να + Present Tense)
να + Present Tense is used to show a prolonged or repeated action or an action in progress.
Μου αρέσει να διαβάζω εφημερίδα κάθε μέρα. = I like to read a newspaper every day. -> repeated action
The verb αρχίζω is always combined with this form and never with the future, since the verb after it indicates an action in progress.
αρχίζω να καταλαβαίνω = I start to understand (understanding is an action in progress)
αρχίζω να κλαίω = I start crying
αρχίζω να πεινάω = I start being hungry
να + Future Form is used to show an instant action.
Θέλω να φύγω. = I want to leave.