First, those texts are two sentences. one is "用餐畢請隨手整理桌面" and another is "座位有限, 請勿休憩佔用". They didn't put punctuation in the two. It's pretty normal in a sign.
2nd, 畢 means “后/後" (after/when ). 用餐畢: after/when you ate food.
3rd, it is not "限請", but "有限", 座位有限, 請勿休憩佔用 means seats are limited, please don't take up after you ate food.
4th, "請": please.
5th, "隨手": about this word, I think the best way to explain is "after/when you ate all food and are going to leave the cafeteria, you clean the table first and then leave please." It's not "you must clean the table" but "at your convenience".

Cassie L.
05/17/21