My question is can 取り組む can be used as a general term for working on something? Like as in "I'm working on something important." 何か重要なことに取り組んでいる。 Or does 取り組む carry too strong a connotation to be used here? Should I just stick with the verb やる as in, 何か重要なことをやっている?
We usually use 取り組む with something that can be called a project or a mission (and such things are normally important, too). But you cannot 取り組む with a task you can finish in less than a day, no matter how important it is. As long as that 重要なこと satisfies this criterion, saying 何か重要なことに取り組んでいる is just fine.
For example, 本の執筆に取り組んでいる is natural, but 重要なメールの返信に取り組んでいる sounds weird.