You leave out the 'become' -- you've already used the verb 'make' so the phrase would be "...make my unproductive time productive." Unless this is a command (it would be a weird one!), it's not really a sentence. You might say "I hope to make my unproductive time productive by turning off my phone." or "This is how I will make my unproductive time productive."

Although I understand what you mean...it just sounds like the words unproductive and productive are used too closely together. In English we try to avoid like-sounding words, even if they mean two different things. This is my suggestion: take out the word either 'unproductive' altogether and say something like: "I'm going to be more productive with my time by multitasking. I'm going to listen to audio books while I clean my house." (The reason I suggest this is because when you say that you plan to be more productive, it gives the listener the assumption that you have already been unproductive.)

Sounds correct to me! You could also say: I"'m going to make my unproductive time productive" Or: "I'm going to make my unproductive time a productive one"