I prefer my hair up, actually.

Literal

Hair [also-も] raised-[side-の方] I [topic-は] like [copula-だ].

An unusual word order where the topic 私は comes *after* the preference clause. More typical order would be 私は髪も上げた方が好きだ. The 髪も ('hair too, also hair') with the も suggests a comparison with something else implicit in context. 上げた方 uses the past form of 上げる ('to raise') in the 方が好き comparative structure ('I prefer the option of having raised it').