Can you say "at least it's sunny" in the sense that finally it's sunny?
Can you say "at least it's sunny" in the sense that it hasn't been sunny for let's say a month but finally it's sunny? Or is it that you can only say "at least it's sunny" when for example you want to say that it's cold and generally everything's wrong but at least it's sunny and for the first case you use "at last" or "finally"?

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At least it's sunny implies that other things are wrong (ie it's windy, it's cold, etc) but at least it's sunny! If it hasn't been sunny for a long time, one would typically say "finally, the sun is out!" Or what u said, "finally it's sunny!"

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Can you say "at least it's sunny" in the sense that finally it's sunny?
No.

Or is it that you can only say "at least it's sunny" when, for example, you want to say that it's cold and generally everything's wrong but at least it's sunny, and for the first case you use "at last" or "finally"?
Yes.