All I'm trying to ask is whether or not my friend is going to be available in the library at the time of lunch(12:00). Even though I'm bilingual/multilingual in three different languages including English, lately some aspects of the language has been confusing to me as I'm learning a new language.
'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.
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'.
I guess there may be a small ambiguity - it could be (unlikely) construed as asking if the person will be lunching at the library. If you want to make it more precise, you could say 'Are you going to be in the library at lunchtime?'
Of course, this is assuming that both you and your interlocutor understand 'lunchtime' to be noon - there could be confusion if your lunchtime is different from theirs!