How to get rid of emotional instability in your partner?

You Can't. Your partner has to want to change --- even then, it's hard. You can share with your partner how the emotional instability has affected you & the relationship. You can also set limits as to what you are willing to tolerate or not.
Therapy. I agree with dr. Copote's answer 100%. I will add that there is a chance your partner never learned emotional regulation/distress tolerance skills. This doesn't mean your partner gets a free pass to act out (remember boundaries). It does mean that individual and/or couples therapy could help by teaching your partner skills to improve your relationship.