I have no idea if I have any allergy to hornet bites but since they are way bigger than those "regular" hornets I suppose a bite or two could be close to lethal?
A single sting will hurt a lot, but won't be lethal to normal healthy adults. Its a different story if you are allergic to bees. It would also be dangerous to receive several stings, but it is pretty unlikely that this would happen to you. In is pretty unlikely that you will encounter a hornet, and even less likely that you will be bitten by one (let alone multiple hornets). Just like with most animals, leave them alone and they'll leave you alone.
In case I or my travel partner would get bitten not knowing if we would react and get into anaphylactic shock at the spot. What should me or my travelling partner do?
Immediately seek medical help. If you have a known bee allergy then you should carry appropriate emergency medication. If you aren't allergic to bees (or don't know if you are) then I wouldn't worry too much as you probably aren't allergic.
I mean if we are in the middle of a forest on a hiking road or something which is not very crowded.
Medical emergencies are a normal danger when hiking. If you have one you should immediately seek medical help. It that means carrying your partner down a mountain then that's what you have to do. It helps to have a cell phone on you, but be aware that it is not a substitute for proper planning. Cell phones often lose service in remote areas.
Does the emergency service speak English? And the emergency room personel? Or perhaps it depends on where I am?
It depends on where you are but they generally don't speak english.
Would regular antihistamine pills help or would that be too late? Even if I put them in at once when I get bitten?
They'll might little depending on how allergic you are, but really if you don't know if you are allergic then you don't need to worry if you are.
Since I have heard of a relative who was nearly killed by a hornet over here I will talk to the doctor about this in case I could be affected the same way. In that case I would probably need to bring one of those adrenaline shot pens or what they are called.
At the very least, talking to a doctor may assuage you of your fears about bee allergies.
Would that be any problem when travelling into Japan, considering the customs and the very strictly regulated procedures of medicine?
You may need to carry a copy of your prescription and might have to declare it at customs (and fill out the necessary paperwork).