Are monopolies always price makers?
You are correct. A monopoly is a price maker. Not a taker.

A monopoly has the power to influence the price it charges as the good it produces does not have perfect substitutes. A price maker within monopolistic competition produces goods that are differentiated in some way from its competitors' products.

I suppose a monopolistic firm could be a price taker via matching random re-sellers prices on singular items, however that would be operating at a loss a majority of the time. The idea with that though is to retain the customer business for the future, not to make a single small profit on one item, one time.