(I'm a U.S. native speaker). I think it's correct, but it could be more natural. I would make two changes. The most important is to change "except" to "besides." The other is to change "have" for "make." "I think everybody should make time for something besides work." "Have" time implies that everyone should simply have a work schedule that is not too heavy. "Make" time implies a conscious decision to manage time and prioritize. I can't explain why I think "besides" is better than "except," beyond saying "it just is."