Is this sentence 100% correct? "Are you going to be in the library at lunch today? "

Solution 1:

Since you are referring to a specific time when you say lunch, I would say "...during lunch?".

Solution 2:

'At lunch' meaning 'at lunchtime' is a usage I was familiar with in grammar school 50 years ago. It's isoformal with 'at break', 'at hometime', 'at Christmas'. And while it could have the locative sense, it defaulted to the temporal (but context was / is important. With some sentences with obvious locatives [Will you be in the library / refectory / cafe ...], the temporal sense is almost forced.

I'll see you at break / lunch / hometime.

We must get together at Christmas.

But contrast

I'll be at lunch in the pub. [non-temporal]

I'll be in the pub at lunch. [most likely temporal, sans other context]

'At' really demands a punctive (non-durative) prepositional object, and 'Christmas', 'lunch', and (more reasonably! ... 15 minutes, as I recall) 'break', and 'hometime', are treated as such here.

Though 'at Christmas' shows that broadening of the punctive into the notionally punctive is often quite acceptable (the duration of 'Christmas' being a movable feast :)), I'd say 'at lunchtime' is more natural sounding than 'at lunch' nowadays. 'During lunchtime' is also perfectly acceptable, but hints at 'at some time during lunchtime'.