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I am able to answer a poisson question where say:
expected value = 2 per week, whats the probability greater than 7 in 3 weeks.

But how would I answer a question where, for example
expected value = 2 per week, whats the probability it will take less than 5 weeks to have more than 13.

Would I need to calculate the probability it takes:
1 week + 2 weeks + ... + 4 weeks


Thanks

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Well first calculate the probability of having 13 or less after 5 weeks as in your first example. And remember $P(\text{not A}) = 1 - P(A)$ ...

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