「Verb in [連用形]{れんようけい} + て + の + Noun」
is a phrase pattern in which the "Verb + て + の" part describes the condition that generates what is expressed by the following noun.
「“[昭和]{しょうわ}な[顔]{かお}”を[買]{か}われての[起用]{きよう}」 means:
"casting based upon his reputation as having the 'Showa-esque face'"
「買われる」 here means "to be regarded highly".
(I am not translating the whole sentence because you did not show us your attempt.)
I am sure some of you have come across the phrase 「[見]{み}てのお[楽]{たの}しみ」, which is in the same structure. "You must see it to enjoy it!"