Should I email the university admin to explain that one of my references is out of office until Monday?
The deadline is still three weeks away, and your letter-writer is back in the office in two business days. Emailing to let them know about a two-day delay would just be noise. In fact, I'd be somewhat surprised if anyone even saw your email response. They're just being polite and helpful. I never got any reminders from my applied-to schools!
Follow up with your letter-writer on Monday, but you shouldn't need to contact the schools you're applying to unless your letter-writer has some sort of personal emergency that means they won't be able to submit on time.