Perhaps you missed the fact that Spock left the information in his computer in such a manner that ONLY his Foster Sister would know how to find it.
Think back to when they first met...
How did he create the alien dragon...
He used his fingers to literally PULL it out of his bedroom computer.
Now think about how Burnham got the info out of the Enterprise computer in Spock's quarters.
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Those are all BIG jumps in logic. That Spock would know that the Enterprise would rendezvous with Discovery, that Burnham would still be on Discovery, that Burnham would want access to his quarters considering they weren't on speaking terms, that Pike would grant said access. Spock is just a creature of habit.
Don't try to explain it, because then I'll start thinking about the holes and it will ruin it.