Your Boss is about to be sacked
First of all, may I express my sincere thanks and gratitude for the great professional tuition and advice that you have been giving me towards my career development?
I have a problem at hand, for which I crave your professional advice once again.
I work as a Finance Assistant in an NGO, with over fifteen years of practical Accounting work experience. I have been working with a Boss who has proved incompetent, and his job is about to be advertised.
What advice can you give me about this situation? Can I apply for the job, or should I exit the Organisation?
Please help me once again with your advice.

This would seem like the perfect opportunity to progress your career.
And I don’t understand why you feel the need to leave the organisation, unless perhaps some of your Boss’s incompetence has reflected on you?
Or perhaps you were party to the incompetence, should have been able to prevent some of it, and so were yourself at least partly responsible. If this is the case you need to sound out opinion about yourself with the relevant management to make your decision. Is the position retrievable?
If not, maybe you should leave, Will you get a decent reference? It’s a small world in professional circles. Maybe stay and repair the damage?
If you don’t feel you were responsible in any way then you should take steps to distance yourself a little from his failures, and talk to the relevant people in management to explain that the shortcomings were not of your making. Be sure to explain that you were just carrying out his orders, and were not involved in any poor decision making.
Are you suitably professionally qualified to take his position? In which case by all means apply.