The usage in that sentence is correct both ways, though it should be written: "We must solve/settle the questions in some way or another". These two words can have the same meaning, like in this case, but settle really means more like "to become still". For example, you could say to someone who is really energetic "settle down", meaning calm down. When used as in the sentence you gave, using "settle" implies making the questions still (metaphorically speaking), or clearing them up. So it has the same meaning in usage but there is a technical difference.

I would also note that instead of writing this in your journal, you should post it in the questions area :)