We don't have any first-hand accounts from the Christian side, but this is not unusual, since they are incredibly rare for much of history and the only one I can think of right now is Procopius from the sixth century concerning Justinian's wars. For the seventh century the sources are very poor and I provided an overview here, but note that none of these sources suggest that they ever fought against the Arabs - instead they were contemporaries who experienced the effects of these wars. There are many, many Muslim sources that attest to earlier events and are described as first-hand accounts that have been passed from one narrator to another. Unfortunately, these accounts were all written down at least a century later so it is debatable to what extent they are representative of what people in the seventh century thought. The story you mentioned seems to belong to this category, but these narrations are difficult to assess. They do contain valuable materials, but they have to be carefully examined and be compared against other sources. In any case, it is unlikely that the Great Library still existed then, since people were already talking about the Library in the past tense in the fourth century, so it seems to me to be a later legend, rather than anything reflective of what happened in the 640s. You will however have to wait for another expert to dig up the source and analyse this narration specifically.

What I can tell you is that modern historians rely on the Chronicle of John of Nikiu to tell the story of the fall of Egypt. John wrote in the late seventh century and is definitely not a contemporary source, but he did seem to have access to some pretty good records and did use a Constantinopolitan source from the 640s to supplement what he found locally; in any case, his chronicle is certainly the first source to describe the fall of Alexandria. An old translation is available here and you can skip down to chapters 119 and 120, which cover the surrender of the city. The story is a fascinating one: first there was a failed Arab attack on the city, then internal dissension within the city between two political factions, before a negotiated surrender to the Arab general. John did have many complaints about the Arabs and the Christian 'apostates' who served in their administration, but they were over taxation, not anything related to the destruction of a library.

For the Arab conquests more generally, it might be useful to read these two answers (here and here), which cover the nature of the wars and should answer some of your questions.