'anti-monarchist' is formed in modern English from the prefix 'anti-' (meaning 'against') combined with 'monarchist' (a supporter or proponent of a monarch).

'monarch' originates from Greek via Late Latin and Old French: Greek 'monárkhēs' (from 'monos' meaning 'single' and 'arkhē' meaning 'rule') → Late Latin 'monarcha' → Old French 'monarque' → Middle English 'monarch' and then 'monarchist' (monarch + -ist). The prefix 'anti-' comes from Greek 'antí' meaning 'against' and has been used in English since classical compounds; 'anti-monarchist' is a modern compound using these elements.

Initially the parts meant 'against' and 'single ruler' (the components of 'anti-' and 'monarch'); over time the compound came to mean specifically 'opposed to monarchy' or 'opposed to monarchs' in modern political usage.