My name in my Japanese passport is different to the one in my philippine passport. I’m going to Japan this June. I’m really worried because if i were to show only my Philippine passport in the immigration booth when departing in the philippines they will ask for a Japanese visa while if I were to show only my Japanese passport, they might think I overstay in the Philippines and maybe ask me to pay for that. On the other hand, if I were to show both, I’m afraid they might question me and in worst case not let me in my flight.
You will need to show a passport on three or four occasions.
On check-in. Here, the agent has to confirm that you meet the requirements to be admitted to Japan and that you are the person on the booking. You want to show the agent your Japanese passport as it confirms you are eligible to enter Japan in the easiest way possible. (In case the romanisation of your name does not match your ticket, you may have to present additional ID with the spelling as on your ticket.) No need for the Filipino passport.
On exiting the Philippines you would want to show your Filipino passport. As a non-Filipino, I am not aware of any exit requirements that may or may not exist; I hope you have your research done. The inspector’s job is not to check whether you are admittable to Japan but only whether you may exit the Philippines. Normally, I expect this procedure to be very rapid. Note that you can typically board any international flight after passing immigration so why should they suspect you’re going to Japan without a visa? Certainly Filipino citizens can go somewhere without a visa.
(Potentially optional) Boarding the plane, you might need to present proof that you are still the same person holding the same ID document that matches your name on the boarding pass. This shouldn’t be more than a quick visual check.
On entering Japan, you could again have come from a number of different (visa-free) destinations on your Japanese passport. So have it ready and show it to the immigration officer. No need for the Filipino one.
When on the return trip (if applicable), continue showing the Japanese passport to Japanese officers and the Filipino one to Filipino officers, except definitely show the check-in agent your Filipino one.
The Philippine authorities are responsible for enforcing their country's immigration rules, not Japan's. Since you're a citizen of their country they should not have a problem with you leaving.
(Actually I seem to vaguely recall that Philippine authorities do have a problem with some of their citizens traveling abroad, in particular young women. That doesn't have anything to do with names in passports, and I'll assume you have researched if you need an exit visa).
As for having different names in the two passports, is it quite well known that Japanese uses a different script from the Latin script used in English or Filipino. Japanese authorities therefore have their own ideas about how to legally write their citizen's names, and there may not be even approximate round-trip identity when they render the Japanese name in Latin script in their passports. So anyone who doubts the Japanese passport it yours should be verifying you by the photo and physical description, date and place of birth, etc.