You are partly right: suo, sua, suoi, sue can all mean his/her's or (polite) your. But the gender of a word like "suo" depends on the noun following it: il suo libro, la sua raccolta, i suoi gatti, le sue macchine. His/her/your book etc.
Shouldn't the possession be regarded to the cook and not to the beer? Otherwise the translation becomes "the cook drinks her beers", not "his".
Could it be possible that the cook could be feminine, or is there a language issue that forces the masculine? Is "la cuoca beve le sue birre" wrong?
@ strahil: Your translation is possible too, but not for your reasons. The possessive gets its form from the noun it belongs to, in this case "birre".
@ TunaSalad: You answer is fine.
"His own beers" is different than "his beers." We need some consistency here!
Your translation "his own beers" is possible if the phrase is said with the stress on "sue". But without the stress on "sue" it can be "his, her or (formal) you".