"He likes beer."
Translation:Του αρέσει η μπύρα.
In case this is helpful, the analogous Spanish construction also requires the definite article: le gusta la cerveza. Whether it means beer in general or the beer he is drinking depends on context. If one wants to stress the latter, then for the sake of clarity le gusta esta cerveza. If the same is the case in Gk, it would be Του αρέσει αυτή η μπύρα. A native Gk speaker can let us know for sure.
Yes, the definite article is needed in Greek even in such cases, as it is the subject of the sentence (sounds like a good rule of thumb, actually). There are cases where the article can be omitted when the respective noun functions as the object of the sentence, but I think it's quite fixed when the noun is the subject of the sentence.
In case it helps, του αρέσει is like le gusta in Spanish or gli piace in Italian -- the subject of αρέσω/gustar/piacere is the beer, and the person is an indirect object (to whom is the beer pleasing?).
That is why Αυτός is not appropriate (just as Él gusta la cerveza/Lui piace la birra would not be correct).
I take it η μπύρα is nominative
is του genitive?
The genitive case forms of the short-form personal pronouns (e.g. μου, σου, του) are used for the indirect object in Greek. The bear appeals "to him".
German has a similar construction: Das Bier gefällt ihm (with the dative, which German still has but Greek lost).